Without Black's h-pawn, the position would be an easy win for White. The simplest way would be to play Ra8, a6, a7 then push the f-pawn to f7. After Black's Kxf7, White plays Rh8, queening his a-pawn or winning the black rook after Rxa7 by a check on h7.
The question is whether Black's h-pawn can run fast enough to distract White from that plan. Here's how the game went: 1.f6 h4 2.Ke6 Re4+ (otherwise 3.Ra8+ and 4.f7 wins) 3.Kf5 Ra4 4.Rg7+ Kf8 5.Rb7 h3! (5...Rxa5? loses to 6.Kg6 Ra8 7.Rh7) 6.Rh7 Ra3 7.a6 Kg8! 8.Rg7+ (Not 8.f7+? Kxh7 9.f8=Q Rf3+) 8...Kf8 9.a7 h2 10.Rh7 Rxa7 and a draw was agreed. After 11.Rxh2 Ra1, White can make no progress.
How much of this did White see when considering his move in the diagram position? He must have got as far as 5.Rb7, but may have been seduced by the pretty winning position after 5...Rxa5 6.Kg6.
Had he thought about 5...h3, he would probably have gone back to the start and looked at the other pawn push, which wins quite easily.
After 1.a6! h4 2.Ra8+ the h-pawn is too slow to create problems. 2...Kf7 loses to 3.a7 h3 (3...Ra5+ 4.Kf4 Ra4+ 5.Kg5 changes nothing) 4.Rh8 h2 (what else?) 5.Rxh2 and now 5...Kg7 6.f6+ or 5...Ra5+ 6.Kd6 Kg7 7.f6+ Kxf6 8.Rh7 with an easy win for White.
Finally 2...Kh7 (2...Kg7 3.f6+ is even worse) 3.f6 also lea- ves the white pawns romping home too fast. In such endings, everything must be worked out at the start. There is no room for guesswork.