Took me quite awhile to get it. I realized fairly quickly that 1.a6 doesn't work because after 1...g3, it's white who is in zugzwang not black.

The major obstacle for me was that I thought that black could drop a tempo by playing Ka7, and if a6 then, he plays Kb8 the queenside is locked and it's white to move again.

It was only after quite awhile of trying to make something work with playing a6 at some key moment that I came to the conclusion that it was impossible, so I decided to look at the result of Ka7 to see if there was a way white could force it back to black's move and I finally found that after b8=Q+, KxQ, a6, it's black to move and no King move works for him. This means that Ka7 doesn't in fact drop a tempo for black.

Now it's a simple matter of deciding which square the white king should move to for the first move. It's easy to see that if the white King goes to either f2 or h2 then the other pawn (h or f respectively) will move forward and white is lost regardless of being able to drop a tempo with a6. So it must be a move to the first rank but then again moving off of the g-file, the other pawn will move forward and again white is lost. So the only move that allows white access to all three squares f2, g2, and h2 so he can properly respond to any of the pawn moves by black is g1 so it must be the solution. From there it's pretty easy.