\begin{align*} \sigma^2 & = E[X(X-1)] + \mu - \mu^2 \\ & = \sum_{x=0}^{\infty} x(x-1) \cdot \frac{\mu^x}{x!} e^{-\mu} + \mu - \mu^2\\ & = e^{-\mu} \mu^2 \sum_{x=2}^{\infty} \frac{\mu^{x-2}}{(x-2)!} + \mu - \mu^2\\ & = e^{-\mu} \mu^2 \sum_{k=0}^{\infty} \frac{\mu^k}{k!} + \mu - \mu^2\\ & = \mu^2 + \mu - \mu^2 \\ & = \mu \end{align*}