Why is a rook worth more than a bishop?

I’ve always wondered why is a rook worth more than a bishop? They do pretty much the same thing, except one moves diagonally and one moves straight, and I don’t believe it’s because of castling. (seems like a stretch) When I played chess before I knew what each piece was worth, I really valued them the same, so....why?

Because a rook can reach any square on the board and the bishop has only access to half the squares. Also if you put a rook on an empty board, it always controls 14 squares, wheareas the bishop only controls 7 to 13 squares.

That is an accepted generalization developed over hundreds of years of trial and error.

https://en.m.wikipedia.org/wiki/Chess_piece_relative_value

If you try to checkmate with KB vs K, you will get an impression of the difference. Or play an endgame KB + Pawns vs KR + Pawns where the opponent puts all his Pawns on the color complex where your Bishop doesn't see them.

Bishop can only go on one color square.

If this was a position, it would be a draw 

No matter how many moves it takes, the black bishop will never be able to capture the white bishop, and vice versa.

A rook can get to all 64 squares, a bishop only to 32 squares.
R+K can checkmate K, B+K cannot, even checkmate with B+B+K  is more difficult.

As others have stated, bishop can't reach half of the board. There is one other reason as well.

At any given moment, rook controls 14 different squares (if we say that neither bishop or a rook is restricted by his own pawns and pieces). If the bishop is placed in the corner, it controls only 7 squares, while in the middle of the board bishop controls 13 squares at the same time.

To sum it up, bishop can't access half of the board and it controls less squares at any given time (from 1 to 7 less squares).

Mmm, okay. I guess a bishop has less mobility, that’s simple enough. Thanks!

all the above are obviously correct...but there's one more =)

rooks a bigger piece in size than a bishop =)

so when the rook gets captured, you apply more energy to lift it and put it to the side than bishop. =)

so rook points is more =)

Rooks and queens have the ability to make impassable lines the king cannot get through. The rook is only weak to an enemy king by the corners. If it's in the corner, where rooks don't belong, you only have 1 square you can safely attack it.

Another important detail is that rooks can deal with any pawn because they can get on the back ranks much faster and don't have to do the shuffle. 

There are plenty situations where a Rook absolutely belongs in a corner.

Like the beginning of the game  

Sometimes it belongs in a corner even if it has a choice

Note that the Rook is the only piece that controls always the same amount of squares, independent of it's position on the board.

yeah idk its just stronger piece generally

Against a King, Rooks present a solid "Line of Death" that the King cannot cross (Cue clip of Monty Python's Black Knight {whose helm looks rather rookish} stating "None Shall Pass".).  A single Bishop leaves gaps (picture the row of bollards at a park entrance that allows pedestrians, but not cars to pass) which need to be closed off with the other Bishop, Queen, or a few well-placed Pawns.  Summary: a lone Rook can hold a King back, but a Bishop needs a support team. 

But by this argument, shouldn't it make the rook "less mobile" than the bishop?

Yea...so it has more weightage, gravitas, and stuff lol =)

also...its harder to capture coz of its weight =)

In terms of the in between pieces (Knights, Rooks, Bishops)
Knights are excellent in closed positions, but cry in open positions.
Bishops will WEEP in closed positions, but are free to do whatever they want on an open board
Rooks are excellent fighters in both. 
Queens are merely premium rooks