This is one of those problems covered in endgame books on corresponding squares.
The win zone for White is if he gets his King to b5, g5, or h5.
The concept of corresponding squares is if one side is trying to win, and the other is trying to draw, if the side trying to win goes to a number, the defending side must be able to go to the corresponding number or he loses.
Since b5 is a winning square for White, The square c4 gets a 1. Since g5/h5 is also a win zone, and it takes 5 moves to get to h4, Black must be able to get to g6 in 5 moves, and so a6 is no good. The king must go to b6, so b6 gets a 1.
Now because d3 is adjacent to c4 (in king move terms) and a step closer to h4, it gets a 2, and Black must go a step closer to g6, so c7 gets a 2.
Now c3 is adjacent to both a 1 and a 2, so it gets a 3. The b7-square is the only place from which Black can access 1 and 2, so it gets a 3.
Now d2 is adjacent to 2 and 3, and is 4 steps from h4, so it gets a 4 as does c8.
Now c2 is adjacent to the 2, 3, and 4, so it gets a 5 as does b8.
Now b3 is also adjacent to 1 and 3, so it also gets a 2, and b2 gets a 4.
The back rank for White gets another 2, 3, and 2. The a-file for Black repeats the c-file, so a7 is a 2 and a8 is a 4.
So now the trick is the fact that no number on either side is adjacent to itself. Black currently sits on a 2. He can get to a 4 (a8), 5 (b8), 3 (b7), or 1 (b6), but he cannot get to a 2.
Therefore, White wins by going to a 2, which is 1.Kb1!
After 1.Kb1, if 1…Ka8, you go 2.Kb2. If Black tries to be coy and never leave the a-file, the 1 becomes the problem. 2…Ka7 3.Kb3 and now 3…Ka8 fails to 4.Kc4 and Black cannot stop Kb5.
If Black goes to the b-file, with a 5 or 3, you go to your 5 or 3 and again, Black will get tempoed out either for the b5-square, or the run to g5/h5.
So the biggest question is what if 1.Kb1 Kb6, since White cannot get to a 1 right away. From the 1, Black can get to 2 or 3, but not 4 or 5, so 2.Kc2 and now what for Black? If 2…Kc7, then 3.Kd3 wins. If 3…Kc8, then 4.Kc4. If 3…Kb6 or 3…Kb7, then 4.Ke2 and White gets to h5.
If 2…Kb7, then 3.Kc3 and now Black has is stuck. 3…Kc7 4.Kd3 already we said is losing. 3…Kb6 4.Kc4 and now White gets to b5 or 4…Ka6 5.Kd3 and he gets to h5.
It all has to do with the corresponding squares and 1.Kb1! Is the only way to do it!
1.Ka2? Kb7 (or Kb8) draws
1.Kb2? Ka8 draws