Guide me on how to solve this puzzle

The first one I solved pretty quickly (10 or 15 seconds?), perhaps because there just weren't that many things to try.  I figured the pin on the g pawn probably played a role, but there just wasn't enough firepower to take advantage of it immediately.  Re1 solves that problem.

The second one was trickier, probably took me a minute or two.  It's insanely clever.  One principle I employed was noticing that white can get a perpetual, so black has to keep calling check if he wants to win.  So I only had to consider moves that call check.  Eventually I noticed the clever idea of mating with the queen on h3, such as ...Bg2+, Rxg2, Ng3+, hxg3, Qf1+, Rg1, Qh3mate.  But white can escape a couple of different ways, such as by Kh2 instead of Rg1, because his moves aren't forced.  But I thought the mate on h3 was so clever it was probably the theme, so I tried the moves in different order until I found an order in which all of white's moves are forced, which is ...Bg2+, Rxg2, Qf1+, Rg1, Ng3+, hxg3, Qh3mate.  Nice puzzle!

There are different puzzle solving strategies out there that deal with this sort of question. Some puzzle solving strategies have you looking for patterns that you can see in a position, like the king and the queen on the same rank in position 1 or the Ng3 double-attack in position 2. I've never liked this approach on its own too much, because there are plenty of tactics where you must first set up the pattern, before you can take advantage of it. For example, a simple puzzle might include you having to check the king first, make it go to another square, which in turn sets up a pin on the diagonal the king has now stepped on to for you to take advantage of.

I think the most reasonable strategy is to look at forcing moves. Forcing moves are putting your opponent into check, threatening checkmate, attacking their insufficiently or undefended pieces or attacking a piece with a higher value than the one attacking it (rook attacking a queen for example) or taking their pieces. A puzzle solving strategy that incorporates this is the 'check, take, attack' solving strategy (that you can also use in your games). This strategy makes you look at all the moves that put your opponent into check, that take pieces or attack pieces (or create mating threats). 

Both these positions don't have a lot of forcing moves to look at. The first one has Nxf3+, Qxg2, Qxf2 (checks), Rxf3, Re1, Nd3 (attacks of insufficiently guarded pieces or higher value pieces) and Ng4 (mate threat). It's quite clear that you have no follow up after Qxg2 or Qxf2, so they don't need much analysis. Ng4 can also just be taken to stop the threat. After Nd3, white just moves the rook to e2, crisis averted. So then you have Nxf3 and Re1 to look at. Then after 1. ... Re1, 2. Qxe1 or 1. ... Nxf3+; 2. Rxf3 the process repeats itself: you'll look at forcing moves. So after 1. ... Nxf3, 2. Rxf3 you might see 2. ... Re1, 3. Rxg3 (again: look for forcing moves) ... 3. Rxf1+; 4. Kxf1 and fxg3. Now your board vision needs to be strong enough to see that in this end position you have no extra material. So you check the other line (maybe returning to this to evaluate the endgame if the other line doesn't give you any material either). After 1. ... Re1, 2. Qxe1 we'll find 2. ... Nxf3+ though and now we can see we win material after either 3. Kf1, Nxe1 or 3. Rxf3, Qxe1+.

The 2nd position is an even better example of looking at forcing moves and to not stop analysing a line too soon. So you start looking at the moves that are forcing and you'll quickly notice that 1. .. Bg2+ and 1. ... Ng3+ seem to be the most promising. After 1. ... Ng3+, 2. Rxg3+, Qf1+; 3. Rg1, Bg2 is checkmate. However, we always need to look at the options of our opponents too and we'll notice that after 1. ... Ng3+ they have another option with 2. hxg3 and we'll analyse this line and find that we have no other forcing ideas than 2. ... Bg2+ which doesn't give us anything after 3. Rxg2, Qf1+; 4. Kh2. So we go back to the other line and we see that after 1. ... Bg2+ we again have 2 forcing moves after 2. Rxg2. Basically 2. ... Ng3 seems to be the same as the line we just looked at, so 2. ... Qf1 is evaluated. 3. Rg1 is the only move. Again we'll look for forcing moves and we'll see 3. ... Ng3+. 4. hxg3 is the only move and then 4. ... Qh3 is checkmate, which we'll find because we'll look at the forcing (check!) moves. Again, we'll need strong enough board vision to see that on move 4 we have a checkmate. 

So, do we need a lot of imagination for these puzzles? I'd say no. It's just the same process over and over again. Look for the forcing moves in the position you've reached. You do, however, need visualisation skills. You need to be able to look ahead and picture in your mind's eye what the exact position is after a few moves. 

Pattern recognition will help you see the forcing moves faster and will help you eliminate some of the other forcing lines a little quicker. You can learn pattern recognition by doing tactical puzzles. The more you do them, the faster you'll begin to notice the patterns. The more practice you get, the better your visualisation skills will become.

Positions where there are a lot of forcing moves are easier to navigate than positions where the moves aren't forcing. To take a metaphor from Kotov's book 'Think like a Grandmaster", imagine that you write the moves down of each lines and make 'trees' out of them. In the second position your solution for example would be a big tree with only a small branch at move 2, because there's also 2. ...Ng3+. For the rest it's straight as an arrow. White doesn't have much choice, but neither does black. Basically if you look at all the moves you'll have a bit of shrubbery from the other candidate forcing moves (like 1. ... Ng3+) that never reach very high and the 1 big tree trunk that is the right answer. Now imagine a position without forcing moves. For example a position where you're trying to look at a potential piece sacrifice to open up their king but don't have a concrete follow-up. Now your tree has all kinds of branches, because at every move there are different possibilities to explore. And there might be yet another tree standing right next to it that needs to be analysed, starting with another strong move that doesn't sacrifice the piece. Suddenly you have a jungle of variations that you need to go through, usually (during the game) in a limited amount of time. That's when things become really difficult and it comes down to how well you can focus, how fast you can analyse, how far you feel you need to calculate in order to get a good evaluation of the move, etc. etc.

Another example might be the endgame, where there just are a ton of moves and you can't analyse everything. That's why in the endgame, where there are trees with so many branches, you do want to imagine positions that you want to reach. Luckier that's also easier to do in most endgames than it is in the middlegame. So, to sum up, in forcing positions you need to have strong visualisation and look for forcing moves using visualisation and go as deep as you need; in positions with a lot of different possibilities, you'll need a lot more imagination and maybe sit back and focus on making a strong plan for the next few moves than trying to work out all the possibilities. Unless of course you're sacrificing a piece, then you'll have to calculate a ton and probably wing it anyway when the position gets way too complicated to calculate all the way through.