Own photograph
Don't fall in love with "natural" looking moves!
There is a chess book titled "Move First, Think Later". Don't take that as a literal advice.
Before continuing please try to find the correct white plan for yourself. It took me several minutes to solve this puzzle in the least efficient way.
I immediately focused on the "natural" move 1.Kf5 to collect h6 seeing the other move 1.Kd4 only as a weird emergency exit. If Black goes for the white a-pawn, then White grabs the black pawn first, gets a queen and blocks the black pawn: 1.Kf5 Kc5 2.Kg6 Kb4 3.Kxh6 Kxa4 4.Kg5 Kb3 5.h6 a4 6.h7 a3 7.h8=Q a2 8.Qa1
If Black stops the h-pawn "conventionally", then White gets a queen: 1.Kf5 Ke7 2.Kg6 Kf8 3.Kxh6 Kg8 4.Kg6! (to prevent Kh7) Kh8 5.Kf5 Kh7 6.Ke5 Kh6 7.Kd5 Kxh5 8.Kc5 Kg6 9.Kb5 Kf7 10.Kxa5 Ke7 11.Kb6 Kd7
White is just in time to play 12.Kb7 preventing Black from playing Kc8.
In a blitz game I probably would have stopped calculation to play 1.Kf5, but I hesitated and finally discovered the problem: 1.Kf5 Ke7 2.Kg6 Ke6 3.Kxh6 Kf6 4.Kh7 Kf7!
Black confines the white king in front of the h-pawn. In order to get out White has to push the pawn to h6. Therefore Black can collect the h-pawn faster and saves the game: 5.h6 Kf8 6.Kg6 Kg8 7.Kf6 Kh7 8.Ke6 Kxh6 9.Kd5 Kg7 10.Kc5 Kf7 11.Kb5 Ke7 12.Kxa5 Kd7 13.Kb6 reaching the second diagram, but it is Black to move: 13...Kc8!
So I was forced to look to at the "counter-intuitive" move 1.Kd4.
There are no disadvantages compared to 1.Kf5:
If Black goes for the h5-pawn, then White grabs the black a-pawn first. Compared to 1.Kf5 Kc5 the a4-pawn needs one more move to queen than h5. But the black h6-pawn also needs one more move to queen than a5.
Black loses the a-pawn anyway due to Zugzwang: 1.Kd4 Kc6 2.Kc4 Kb6 3.Kd5 
The difference is, that White later progresses faster on the kingside due to the more advanced pawn: after Kxh6 White immediately threatens Kg7. The black king has to collect the white a-pawn first and is too slow to reach f7 or f8.
Here is the position to replay:
Take-away
If you have two decent moves, don't calculate the more "natural" looking option like crazy, if the other move doesn't have obvious disadvantages - easier said than done!
