By definition, $e^x : = \sum_{n = 0}^{\infty} x^n / n !$ for all real numbers $x \in \mathbb{R}$. Thus, applying
R1814: Expectation of discrete random euclidean real number, we have
\begin{equation}
\begin{split}
\mathbb{E}(X)
& = \sum_{n = 0}^{\infty} n \mathbb{P}(X = n) \\
& = \sum_{n = 1}^{\infty} n \mathbb{P}(X = n) \\
& = e^{- \theta} \sum_{n = 1}^{\infty} n \frac{\theta^n}{n!} \\
& = \theta e^{- \theta} \sum_{n = 1}^{\infty} \frac{\theta^{n - 1}}{(n - 1)!} \\
& = \theta e^{- \theta} \sum_{n = 0}^{\infty} \frac{\theta^n}{n!} \\
& = \theta e^{- \theta} e^{\theta} \\
& = \theta
\end{split}
\end{equation}
$\square$