Well I believe the basic idea is you are exchanging a side pawn for a center pawn. Also with the c-pawn out of the way, black can develop the Queen along that diagonal and also get the rook along the semi-open c-file. I'm not 100% sure though, that's just how it seems.
Why exchange c-pawn and d-pawn in Sicilan

After cxd4, black has 2 central pawns (versus whites one) and a half-open c file. The negative side is that black gets a bit behind in development.

Note that black also usually more or less has to exchange after white plays d4. If white goes for d4 he will in many variations of open sicilian have the more active pieces (for example knight d4 is very active) and development advantage (black has played the non developping c5 and often will also soon play moves like a6) in the early part of the game. Also, altough black potentially has more pawns on the central files, currently the white pawn on e4 is the only pawn strictly in the centre. Thus black will have to proceed very carefully in the early part of the game because after the slightest inaccuracy he might just get blown off the board.
To compensate for this black in many sicilian lines has a very healty position without any major weaknesses plus his pawn majority in the centre plus some other long term plusses. Blacks position has considerable counterattacking potential and sicilian endgames are often excellent for black. Because of blacks good long term prospects white is practically forced to attack and try to chekmate black or, if this is not possible, at least to force some concessions. In short, sicilian is very unbalancing opening designed for fighting.

After cxd4, black has 2 central pawns (versus whites one) and a half-open c file. The negative side is that black gets a bit behind in development.
Black is no more behind in development than he would be if he didnt take the pawn. The knight moving from f3 to d4 is just the same piece moving twice, its not a developing move, so black is still the same amount behind in development that he was before taking this pawn.
But Black is usually still lagging in development, especially on his Kingside. Black has to move another pawn to develop the dark Bishop, where White has already played e4 and d4, and is free to attack all-out. 1. e4 obviously opens a lot of lines for White, while the c5 move only helps Black in the longterm; the pawn move itself contributes nothing to opening development, unless you want to run out the Queen! 1. ... c5 puts pressure on d4, but after you exchange pawns, you give up that influence in the center. But as thejackbauer mentioned, you get the semi-open file for the middle game.

The extra pawn in the center helps, but I have a lot of problems trying to achieve the d5 break. It's even less fun when you have a backwards pawn on d6 :(
avocado_black
The idea is that by accepting the center pawn, you give White an early attack, but you gain longterm advantages in that you will have a better pawn structure, which will favor you in the endgame.
Often times, Black creates a "mini-center" with e6 and d6; while the position is somewhat cramped, the center pawns make it very difficult for White to muscle through right away, buying Black some time to catch-up in development. And you still have the semi-open c-file to generate an attack of your own.
When you play the Sicilian, you're issuing a challenge to White: he gets the early attack, but after every turn, you move closer and closer to an endgame in which you have the advantage. White has to prove that he can make use of the tempo you give him. And you have to prove that you can make better use of the extra pawn in the center.

(1) It is white's choice to play 3.d4 (Open Sicilian). Without it we have the Closed Sicilian which gives black an easy game.
(2) Black''s plays 3...cxd4 to eliminate one of white's important center pawns; black has "an open c-file for counter-play, along with his two center pawns that play their initially defensive role" - (MCO14)
Suppose black plays 3...Nf6 (the next commonest move):
3...Nf6 4. dxc5 Nxe4 5. cxd6 Nxd6 is one line.
The commonest white line after 3...Nf6 is 4. Nc3 (white now really threatens dxc5) where again by far the commonest black move is 4...cxd4. If black does not play that now, there are several rare lines (4...Qa5, 4...Nc6, 4...a6, etc.) in which white gets the better devcelopment, position or the center.
I don't know if you can call the Closed Sicilian an "easy game" for Black. I think Modern GMs laugh at things like the Grand Prix Attack, but it looks pretty good in that book "Play 1. e4" or something. I would think that some pros still include it in their repertoires.

I don't know if you can call the Closed Sicilian an "easy game" for Black. I think Modern GMs laugh at things like the Grand Prix Attack, but it looks pretty good in that book "Play 1. e4" or something. I would think that some pros still include it in their repertoires.
Books that advocate lines like the grand-prix or closed sicilian for white tend to omit the critical lines. They do this to make the opening seem better than it is, and therefore sell more books. Roman Dzindzichashvili is famous for doing this sort of thing.
Speaking of the grand-prix, Taimanov sicilian players might want to take note of a line I like to use against the mainline.

This exchange greatly benefits black's long term position.
The reason is that now black has 2 pawns in the center, versus white's single pawn. SInce blacks pawns outnumber white's in the center, white cannot smash through the center with pawns, he would have to sacrifice a piece for that(and sometimes he does). White can position aggresively in the center, but black's pawn will usually be able to fend off the qhite peices by depriving them of access of critical central squares.
At some point, when black has completed his development for example, black can use the extra center pawn to his advantage, by pushing the center pawns forward, cramping white.
You will notice that in many open sicilian games white's focus will shift from the center, to one of the flanks. This is because blacks pawns have suceeded in making it difficult for white to make any progress there. However, if white commits too much toawrds some flank, black can make a strong counterattack in the center..its a pretty good and well known rule of thumb that an attack on a flank should be countered by an attack in the center.
As a sicilian palyer you would want to take care that white cannot paralyze your center pawns without commiting some pieces to continue that work. If he is able to paralyze these pawns with minimal effort then he can concentrate on a flank very profitably, as your center pawns not only are deprived of their thrusting counterplay, they also act to cramp your own pieces.
You guys speak as if Black has an advantage in the Scicilian which is not the case in any opening. Every opening struggle is a fight for black to be able to equalize or make the position unclear. You can't just tell someone "blacks pluses" and end it there if you want to make a dececent thematic description of an opening you have to show what both sides are planning no matter what side you are intending to play. That being said I am far from an expert at the Scicilian for any side but I know that at the very least practice has shown that these magical "pluses" in black's position while they are there ,they are not as absolute as you make them sound. avocado_black if you want to learn the Scicilian you gotta put in alot of work...learning any opening for the black pieces is always harder than for white and there is no opening in chess that will outomatically give black equality- much less advantage. So I suggest that if you want to learn an opening you start by knowing theory for both sides and treating the opening with the due respect.

Thus black will have to proceed very carefully in the early part of the game because after the slightest inaccuracy he might just get blown off the board.
In short, sicilian is very unbalancing opening designed for fighting.
I have tried to play the Sicilian although I do not have time to study...this does not work. The Sicilian requires "a lot" of study time"...I guess just like most openings. I have had some success with it although I have had some games that ended quickly, and not in my favor. If you do not know how to play the Sicilian properly, you can lose very quickly as I have learned myself.
I have to agree that the Sicilian is very unbalancing and thus a fighting opening. That is why I am not thrilled to see it played against me. Anyway, if you want to "study" it, make sure you have plenty of time and material, because you will need it. I might be heading back to my old Scandinavian Defense because of my lack of study time; the Sicilian is just an incredible amount of theory.

Sicilian basics:
Black gets 2 center pawns
Black gets compound pawn structure (less pawn "islands" i.e. the dragon)
White gets piece activity early
Black has to force counterplay (often with a rook exchange-sac on the cfile)

By pawn-islands I was speaking comparatively to other openings, not comparatively to white. But I guess when I look at it that way there wasn't really a reason to mention it.... Regardless, you still get a good compact pawn structure

Ohhh, thank you for your wonderful answers everyone!
I didn't expect to get so much information, so I'm printing these out and begin to read now.
Thank you very much!

About the advantage of the central pawns, post #2 is spot on. But of course the other benefit is the open c file for black, which is another key part of black's strategy, what gives him the queenside counterplay. Black certainly gets many things, but they are long term and do allow white some sharp attacks sometimes.
Black's center is flexible. It can be preventive with e6 and d6, or aggressive with e5 and d6, if he can control the d5 hole well enough.

This exchange greatly benefits black's long term position.
The reason is that now black has 2 pawns in the center, versus white's single pawn. SInce blacks pawns outnumber white's in the center, white cannot smash through the center with pawns, he would have to sacrifice a piece for that(and sometimes he does). White can position aggresively in the center, but black's pawn will usually be able to fend off the qhite peices by depriving them of access of critical central squares.
At some point, when black has completed his development for example, black can use the extra center pawn to his advantage, by pushing the center pawns forward, cramping white.
You will notice that in many open sicilian games white's focus will shift from the center, to one of the flanks. This is because blacks pawns have suceeded in making it difficult for white to make any progress there. However, if white commits too much toawrds some flank, black can make a strong counterattack in the center..its a pretty good and well known rule of thumb that an attack on a flank should be countered by an attack in the center.
As a sicilian palyer you would want to take care that white cannot paralyze your center pawns without commiting some pieces to continue that work. If he is able to paralyze these pawns with minimal effort then he can concentrate on a flank very profitably, as your center pawns not only are deprived of their thrusting counterplay, they also act to cramp your own pieces.
You guys speak as if Black has an advantage in the Scicilian which is not the case in any opening. Every opening struggle is a fight for black to be able to equalize or make the position unclear. You can't just tell someone "blacks pluses" and end it there if you want to make a dececent thematic description of an opening you have to show what both sides are planning no matter what side you are intending to play. That being said I am far from an expert at the Scicilian for any side but I know that at the very least practice has shown that these magical "pluses" in black's position while they are there ,they are not as absolute as you make them sound. avocado_black if you want to learn the Scicilian you gotta put in alot of work...learning any opening for the black pieces is always harder than for white and there is no opening in chess that will outomatically give black equality- much less advantage. So I suggest that if you want to learn an opening you start by knowing theory for both sides and treating the opening with the due respect.
Zxb, he's just mentioing black's plusses, but I guess to less experienced players it may give them the wrong impression. White is getting lots of development and, at the moment, has more space in the center which can certainly be dangerous for black.
I don't think fiveofswords was praising black so much as he was justifying black's strategy of playing ...cxd4. I don't think he implied that these things are easy to achieve and that white doesn't have any counter plans. His post was indeed an incomplete thematic description of the opening, but it was only meant to describe black's plan as that's what the OP is asking.

Well I think every White line, except maybe stuff like the Smith-Morra, has an advantage in the opening phase, as many people have said. I doubt anyone can just force equality. Black has to make a lot of annoying pawn moves to buy him time, while White can go full-steam ahead in developing pieces for an attack, giving him an advantage for the time being. But there is a lot of time in the middle game and endgame for both sides to outplay the other.

For some reason, I don't like the alapin (as white, because I like to play against it as black). I guess it's just too slow, clumsy, and dull for me. White's ambitions to get a nice center for nothing definitley doesn't happen, and I don't like the resulting positions that come out of it. And I never say dull lightly, as many people think the french is dull yet it's my favorite opening.
I just don't think losing a central majority is anything to be scared about (except the amount theory that comes from the open) and that it's a lot easier for white to get play in the open than the alapin. True, the alapin makes it harder for both players, but if I wanted a quieter struggle that has less theory, I would much rather play the closed. I like the closed quite a bit. I don't play e4 anymore though so I don't have to worry about the damn sicilian

If you want a quieter struggle why not play lines of the open with an early g3? Very positional, and with a lot more theoretical bite than a closed or alapin.

Well for me, it's just like, why not just put the bishop on e2 and on f3 later, but I never looked into that.

Nigel Davies wrote a book on this way of playing some years ago.
http://www.amazon.com/Taming-Sicilian-Repertoire-Against-Popular/dp/1857443012
In my opinion, an excellent positional sicilian repertoire would be the g3 systems against the e6 and d6 sicilians, and the Rossolimo vs. the Nc6 stuff. The Rossolimo is by far the best anti-sicilian, and you can't really get the g3 stuff to work against the Sveshnikov/Kalashnikov setups.
I'm trying to make Sicilian one of my weapons as Black now.
And I have one question which would be very easy
for the one who often plays Sicilian...
1 e4 c5
2 Nf3 d6
3 d4 cxd4
is very usual line in Sicilan I guess.
Here (and also in many lines), Black exchanges his c-pawn and the opponent's d-pawn.
But what is the benefit of this for Black in terms of his whole strategy ?
Black desires to exchange them with his will ? or
White forces Black to exchange them and Black just have to do it ?
I would be grateful if you know and teach me...
Thank you for your time :)