Intuitively, it's not hard to believe that for a ring $R$, the matrix ring $M_{mn}(R)$ is isomorphic to $M_m(M_n(R))$. Taking a matrix in $M_{mn}(R)$ and turning the $n\times n$ blocks into single entries would give a matrix in $M_m(M_n(R))$, and taking a matrix in $M_m(M_n(R))$ and "erasing" the brackets of each entry would give something that looks like it belongs to $M_{mn}(R)$.
This is embarrassingly informal though, what's the proper way to show they are isomorphic?