Since $I_E$ takes values in $\{ 0, 1 \}$, then $e^{i t I_E}$ takes values in $\{ e^{i t \cdot 1}, e^{i t \cdot 0} \} = \{ e^{i t}, 1 \}$. Applying results
one then has
\begin{equation}
\begin{split}
\mathbb{E}(e^{i t I_E})
& = e^{i t \cdot 1} \mathbb{P}(I_E = 1) + e^{i t \cdot 0} \mathbb{P}(I_E = 0) \\
& = e^{i t \cdot 1} \mathbb{P}(E) + 1 - \mathbb{P}(E) \\
\end{split}
\end{equation}
$\square$