Understanding a proof of the Davenport-Rado-Mirsky-Newman theorem
That equation states that $A$ is an exact cover. On the left, you have all the terms $z^m$ where $m$ satisfies $m\equiv a_s\pmod{n_s}$, with coefficient the number of such congruences $m$ satisfies. On the right, you have $z^n$ for every $n$, with coefficient $1$. So, the equality is saying that every non-negative integer satisfies exactly one congruence.