\begin{align*} M(t) & = \sum_{u=0}^{\infty} e^{tu} \binom{u+r - 1}{r-1}(1-p)^{u}p^r\\ & = p^r \sum_{u=0}^{\infty} \binom{u + r - 1}{r-1}(e^t(1-p))^u \\ & = \frac{ p^{r}}{(1 - e^t(1-p))^r} \sum_{u=0}^{\infty} \binom{u + r - 1}{r-1}(1 - e^t(1-p))^r (e^t(1-p))^u \\ & = \frac{p^{r}}{(1 - e^t(1-p))^r} \end{align*}