They just changed the order of summation. Draw a lattice of the first quadrant : i.e. draw $$ \{ (m,n) \, | \, m,n\ge 0 \}. $$ On the LHS, the sum over $m$ is a sum over an horizontal line, and then the sum over $n$ sums over all those lines. On the RHS, the sum that goes from $n = 0$ to $p$ is a sum over the diagonals $D_p = \{ (m,n) \, | \, m+n = p \}$ so that given $n$, you have $m = p-n$. (The function $f(x) = p-x$ is a diagonal that intersects the $x$ and $y$ axis at positive values.) Letting $n$ from $0$ to $p$ means $m$ goes from $p$ to $0$ (in the other direction) and summing over $p$ is just summing over all diagonals.
There is still the problem of convergence, but I can't look at this since the arguments you took were $[\dots]$ and I don't know what that is. =P
Hope that helps,