Consider if it were true that, for any set of 3 horses, they were all the same color (within the set of three). This would mean it would be impossible to have

Consider if it were true that, for any set of 3 horses, they were all the same color (within the set of three). This would mean it would be impossible to have a set of 3 brown horses and a white horse hanging around. Why? Because if you added the white horse to the group and removed another horse, then you wouldn't have a group of three of the same color. So this would contradict our assumption. So, if our assumption at top is true, it can only be true if all horses are the same color. That's all that the original proof is saying, only stated slightly differently. It's saying that the only way the hypothesis could be true for N is if it's also true for N + 1.