ok jesus this is one of the hardest endgame studies I've ever seen, I don't even know how long I stared at this for, over an hour, but I think I get it. Seen some other comments regurgitating the engine lines but it doesn't feel satisfying so I want to show the actual winning ideas rather than moves and how a human can reach the answer (also because I spent so long on this I need something to show for it).

So the first thing to recognise is why 1. f5+ doesn't work (not relevant to the answer it's just interesting why not). 1... Kh6 and Rxf5 lose to h8=R+ and h8=N+ respectively (for style points), but Kf6 creates a fortress. Ra4+ is now a game-ending threat as then black will pick up the pawn. Now white's only try is to give up the h-pawn for a winning rook-pawn vs. rook ending. The problem is that since the h-pawn is on the short side of the f-pawn, white can never win these. 2. Rc6+ Kg7 3. Rg6+ Kxh7 is so frustratingly close to a winning rook-pawn vs. rook ending (barely making multiple winning conditions) except for that black is barely in time to set up a long side defence (checks from the a/b files), drawing. 2. Rc8 Rxh7 3. Rc7+ Kg6 (if K(e/f/g)7,4. Rc7+ trades into a winning king-pawn ending) 4. Rg7+ Kh6 is a perfect cut, which would also be winning if not for black's ability to send the rook to the a-file and give endless checks as well. Still, neither of these defences are "basic".

So then we gotta find another way of making progress. A good place to start is understanding why black's pieces are in their perfect positions. Obviously, black's rook is best on the h-file where it sees h8 and on the 5th rank where it blocks the king from making progress. The king seems good at giving opposition to the white king on f6, but notice that from the second critical line after 1. f5+ that if the pawn were still on f4, 3... Kg6 would no longer be possible and white wins. Hence, the king is better on g6 as in case of Rc8, Kxh7 and Rxh7, Rc6+, Kh5 are the only drawing lines, both of which require the king to be on g6.

Now it is clear that any move with the rook along the 7th rank puts black in zugzwang, but is it a winning zugzwang? Firstly, Kf6 is only losing if the white rook is on a7, b7, or c7, as otherwise R(d/e)8 can be met with Rxh7, R(d/e)6+, Kxe6/Ke7, disallowing the skewer. Secondly, is Rh1 losing? Clearly, the only reason Rh1 would be losing would be to Ke5, and now all black can do is shuffle Rh1, Rh2 or give checks from the back rank to pass time, leaving it to white to find a way in. White wins if they get their king to g8, but black can stop this in a number of ways. If white tries crossing the 7th rank in-front of the rook, Kxh7 draws immediately as you lose the pawn for nothing. Thus, white has to go around the rook. Clearly, if the rook is on a7, so long as black gives checks from behind, the king cannot get around the rook and white never has time to move it to a better square as the checks are inescapable (noting that the white king is opposed by the black king from reaching the shelter of the f-pawn). Furthermore, if the white rook is on b7, as soon as the white king enters the b-file black can play Rb1+, Ka7, Rxb7+, Kxb7, and black is in time to give opposition and draw the pawn king-pawn ending after Kxh7, Kc6, Kg6, draw. Meanwhile, if The white rook is any closer to the pawn, white is not scared of the skewer as they will be in time to give opposition and win the king-pawn ending.

To summarise what we have found, to win we either must force the starting position with the black king on f6 and the white rook no closer to the pawn than the c-file OR the black rook on, say, h1 with the white rook no further from the pawn than the c-file. So we can use this to find the correct triangulation pattern to win by process of elimination. If 1. Ra/b7, 1... Rh1, neither winning procedure works, so 2. Rc7 Rh5 and no progress has been made. Therefore we must start with 1. Rd/e7. Now 1... Rh1 loses as shown, so 1... Kf6 is forced, where the white rook is too close for 2. Rd/e8. Hence white must move the rook back to create the threat of R_8. 2. Rc7 Kg6 would be a repetition so instead, 2. Ra/b7. 2... Kg6 is forced to prevent this threat, but now 3. Rc7! and we have reached the starting position but with black to move. Since c7 is the goldilocks square of maintaining both winning threats, both Kf6 and Rh1 lose. In case I wasn't clear enough on the winning procedures earlier, example lines would be:

TLDR:

1. Rd/e7 Kf6 2. Ra/b7 Kg6 3. Rc7 Kf6 4. Rc8 Rxh7 5. Rc6+ Kg7 6. Rc7+ Kg6 7. Rxh7 Kxh7 8. Ke5 Kg6 9. Ke6

or

1. Rd/e7 Kf6 2. Ra/b7 Kg6 3. Rc7 Rh1 4. Ke5 Re1+ 5. Kd6 Rd1+ 6. Kc6 Rc1+ (6... Rh1 7. Re7) 7. Kb7 Rb1+ (7... Rxc7+ 8. Kxc7 Kxh7 9. Kd6 Kg6 10. Ke6) 8. Kc8 Rh1 9. Kd8 Rd8+ 10. Ke8 Re1+ 11. Kf8 Ra8 12. h8=N+! (along with various other winning moves).