Explanation of the Wolstenholme theorem proof

Question

I recently came accross the Wolstenholme theorem which says that $$\binom{ap}{bp} \equiv \binom{a}{b} \pmod {p^{3}}$$ On wikipedia, it gives a combinatorial proof of this theorem that involves splitting a set A of size ap into a rings of size p, but I don't understand the rest of the proof. Can someone help breakdown the group theory in that proof? I'm familiar with what a direct sum and what the cyclic group is, but I don't understand this proof.