I don't know if this counts as a combinatorial interpretation, but the identity can be seen as follows: Firstly it's equivalent to 
$$m\sum_{k=0}^m a_k c_{m-k} = (n+1)\sum_{k=0}^mka_kc_{m-k}\hspace{40pt}(1)$$
If we write $f(x) = \sum_{k=0}^\infty a_kx^k$ then the LHS resp. RHS of $(1)$ is just the $(m-1)$-th coefficient of the LHS resp. RHS of 
$$(f^{n+1})'=(n+1)f^nf'.\hspace{50pt}$$