I have been trying to understand the reasoning behind the following change of summation but still don't get it down, could anyone explain me what is happening?

The summation is $$\sum\_{n=0}^{\infty}\frac{z^n}{n!}\sum\_{r=0}^n s(n,r)x^r = \sum\_{r=0}^{\infty} x^r \sum\_{n=r}^{\infty} s(n,r) \frac{z^n}{n!}$$

where $s(n,r)$ is the Stirling number of the first kind.