Let's go over it more slowly (and slightly different from but still conform to the explanation by SG), and hopefully without errors.
The key squares are b5, g5 and h5. If white can move onto one of these, he wins, no matter where black's king stands. So, black has to prevent that.
1) White's access to b5 is c4. If white plays Kc4, then black must be able to play Kb6. Why not Ka6? Because then black is too slow to follow white to run to g5 or h5, which is not the case from b6. So, c4 and b6 is the first set of sister squares.
Similarly, white's access to g5 or h5 is via h4. If black can answer Kh4 with Kg6, then he prevents that, making h4 and g6 also sister squares. This plays a key role in counting the moves in the cases where a race is important.
2) Second step. White's access to c4 can be b3, c3 or d3. Black's access to b6 is from a6, a7, b7 or c7. But which one is good in which case?
2.1) if white is on b3, then black must not be on a6. Because white would play Kc2, threatening to go to h4. Black can try:
- Kb7 but after Kc3 Kc7 (squares on the 8th rank are two steps from b6 and after Kb6 white plays Kc4, forcing black off the b6-square) Kd3!, black cannot do both, follow white to the kingside and stay in touch with b6.
- or Kb6. But then white plays Kd2 Kc7 (forced for the race) Kd3 with same position as above
Also, if white is on b3, then black must not be on b7 because then white plays Kc3 and black is forced to play Kc7 (the 8th rank is two steps away from b6, and after Kb6 Kc4 would force black off the needed b6 square) and is left with the same problem as above after Kd3, he can't follow to the kingside AND be in contact with b6.
So b3 and c7 (and a7 but not needed later in the reasoning) are sister squares.
2.2) if white is on c3, then we can eliminate a6 and a7 (too late for the king side race) and c7, because of Kd3 as above. So, c3 and b7 are sister squares
2.3) if white is on d3, then black must not be on a6 or a7 (too late for the kingside run), nor on b7, because then Kc3 forces black off the sister square square b7, nor on b6, because Kc4 forces black off b6. So, c7 is left as the only sister square for d3 as well (it is also a sister for b3 as said above).
3) White can reach b3 from a2, b2, c2; c3 from b2, c2, d2; and d3 from c2, d2, e2
We only need two cases for b3 to find the solution
3.1) if white is on c2, then black must be in contact with a7 or c7 (to match Kb3 next), as well as b7 (to match Kc3 next) and again c7 (to match Kd3 next). Candidates: a8, b8, c8, a6, b6. A8 and a6 are eliminated because of the distance to the kingside. But b6 isn't good either because of Kd2 Kc7 (to follow towards the kingside) Kd3! and black has to leave the sister square c7 instead of going onto it. Similarly, c8 isn't good either because when white then plays Kd2 black has a problem:
-Kc7? is matched by Kd3 forcing black off the sister square c7;
-Kb7? and Kb8? are too slow in the race;
-Kd8? and Kd7? fail over Kc3 Kc7 (c8 is two steps from b6) Kd3!
So, the only sister square for c2 is b8.
3.2) if white is on b2, then black must be in contact with a7 or c7 (to match Kb3), as well as b7 (to match Kc3) as well as b8 (to match Kc2), leaving c8 as sister square for b2.
3.3) we don't need to consider a2 since we have enough material to find the solution in 4)
4) finally white can reach both c2 (3.1) and b2 (3.2) from b1. After Kb1, black Black must move from a7 to a square that leaves him in contact with b8 (to match Kc2), c8 (to match Kb2). leaving Kb7 as the only option. But then white plays Kc1! and all black's moves fail:
- Ka8, Ka7 and Ka6 fail to the kingside race
- Kb8 fails to Kc2 forcing the king off the sister square
- Kb6 fails to Kd2 Kc7 (the race) Kd3 see above
- Kc8 fails to Kd2 and now Kd7 or Kd8 fail on Kc3 Kc7 Kd3! and Kc7 fails on Kd3
The key in all this are the little triangles white's king can make and black can't mirror.
Hopefully this helps. If you If you find errors, let us know so we can try to correct.