Here is a simple proof.

Suppose that $p \in \mathbb{Z}$ $\nmid$ $m + i$ or $m - i$ (as it does not divide their imaginary parts), but that $p \mid m^2 + 1$. Then we know that $p$ cannot be prime in the Gaussian integers. Thus, there exists a nontrivial factorization of $p$ in the Gaussian integers, which can have only two factors (since the norm is multiplicative, and $p^2 = N(p)$), there are at most $2$ factors of $p$, so it must be of the form $$p = (x + yi)(x - yi)$$ for some $x,\:y \in \mathbb{Z}$. Therefore, we have that $p = x^2 + y^2$.

If you are interested, you can see a collection of proofs here.