Help on John Nunn's Puzzle

I've been playing chess casually with friends and I decided to try some puzzles from a book. I found John Nunn's Chess Puzzle Book (2009) but I don't understand puzzle number one. Sorry if I'm beginner but I'm curious if there's something wrong with the puzzle. The provided solution is confusing. If someone can enlighten me just make this clear.

So starting positions are:

White:

• Rook on a7
• King on d4
• Pawn on f5

Black:

• Rook on h7
• King on h6
• Pawn on f6

Then this puzzle says

White to play: The game continues

1 Ra2 Kg5 2 Rg2+ Kxf5 3 Rf2+ Ke6 4 Ke4 Rh4+ 5 Kf3 Ra4

and eventually ended in a draw. Did either side miss something in the sequence?

When i jumped to solution, it says:

3 Kd6! Not 3 Ke6?? Kg5 and it is Black who wins. 3... Kh5 Or 3... Kg5 4 Ke6. 4 Ke7 Kg5 5 Ke6 and wins. It is surprising that White (rated 2247) missed this win.

What i don't understand is how white king jumps to d6 on move 3? Or maybe Im not reading the solution properly?

I'm not sure if im missing something here. Can anyone explain this? Feel free to recommend me a different book if this is too hard for a beginner, but im really curious why the solution is correct.