I think this puzzle is wrong.

The answer given was 1... Ra7 2. Re1 Kf1 3. Re8 Rxe8 4. Rxe8 Kxe8 5. c8=Q

How is 2... Kf1 not a blunder? Why doesn't black play 2... Ra8?

Edit: I couldn't post the board correctly because this is too unintuitive for me. The puzzle starts after black moves Ra7 and it's white to move.

I think this is right.

Any move by black is still losing, but you only need to worry about the side that the tactic is for. The first move should be the same, no matter what the opposing side does.

Some tactics are designed to show specific ideas or patterns and some come from real games where that was the line played.

But why Kf1, though?

You mean Kf8? In a puzzle, the opponent doesn't need to make the best move, the only requirement for it to be a good puzzle is that the correct answer requires us to make the best move (not the opponent). Kf8 is just there to test if we understand why it's a blunder and how we can punish it.

Probably because Kf8 is the only move that allows a tactical continuation and shows why the previous Ra7 didn't work.

But in the analysis, the engine said Ra8 was better. :/

Again, you do not need to worry about what the other side plays. The first move you make i should bbr the best move in the position and rthe replies by the other side may or may not be best play or give the hardest resistance.

Had black played Ra8 instead, there are multiple good moves for white but the puzzle gives a tactical line that is concrete. 

2...Ra8 3. Rd7 with a 4. Re7 idea and Black is almost busted.