Black gives away his rook here because that is the only way to prevent mate on white's next move.

After

1. R-f1

any move by black *other* than ... R-d4

results in immediate checkmate for white.

Sacrificing the rook is the *only* way for black to prolong the game even one more move. So, the puzzle isn't stupid or ridiculous. But I certainly understand the frustration of not being able to understand a move that seems to make no sense when someone is telling me it's the optimal move. I've been there many times!  

PS you can see all this for yourself by walking through the solution and exploring the various move trees given to the right of the board once you've solved it.

1.Kg1, my first thought, doesn't work cos of 1...h5! 1.Rf1! not only blocks the f2 square, but threatens a second mate with 2.Rf3.

If Rg5 then Rf3#