\begin{align*} M(t) & = \sum_{x=0}^{\infty} e^{tx} \frac{\mu^x e^{-\mu}}{x!}\\ & = \sum_{x=0}^{\infty} \frac{\left (\mu e^t \right )^x e^{-\mu e^t} e^{\mu e^t} e^{-\mu} }{x!}\\ & = e^{\mu e^t} e^{-\mu} \sum_{x=0}^{\infty} \frac{\left (\mu e^t \right )^x e^{-\mu e^t} }{x!}\\ & = e^{\mu \left( e^t - 1 \right )} \sum_{x=0}^{\infty} \frac{\left (\mu e^t \right )^x e^{-\mu e^t} }{x!}\\ & = e^{\mu \left( e^t - 1 \right )}, \end{align*}