To reformulate we need to show that the number of partitions of numbers divisible by $d$ into $n$ parts from $0$ to $n-1$ equals the number of partitions of numbers divisible by $d$ into $n-1$ parts from $0$ to $n$ (this is done after we remove one $n$). We can actually construct a bijection which is well-known Ferrer Diagrams: represent the partition graphically on the lattice : firstly put the largest part, then next put the second largest part and so on. The bijection is the rotation by 90 deg. Or, if you want it more algebrically,if we have $p_n$ parts equal to $n$, $p_{n-1}$ parts equal ro $n-1$ and so on, we associate the sysetm $p_n, p_n+p_{n-1}, p_{n}+p_{n-1}+p_{n-1}, \ldots$

Another solution based on generating function

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This also appeared on https://artofproblemsolving.com/community/c6h1520134p9154743 .