This is proposition 12.3.2 from Ireland and Rosen's "A Classical Introduction to Modern Number Theory". $D$ is the ring of algebraic integers of some algebraic number field, and $e$ is the ramification index of $P$. Why does $P^e\subset (\alpha) + P^e$ imply that $(\alpha) + P^e$ is a power of $P$?
In a Dedekind domain, "to contain is to divide"; i.e., for ideals $I, J$ of a Dedekind domain $D$, if $I \subset J$, $J \mid I$. Furthermore, there is unique factorization of ideals in a Dedekind domain. Since the ring of integers $\mathcal{O}_K$ of a number field $K$ is a Dedekind domain, both of these facts hold for your ring $D$. So, because $P$ is assumed to be a prime ideal, $P^e \subset (\alpha) + P^e$ implies $P^e + (\alpha) \mid P^e$ and by uniqueness of prime decomposition, $P^e + (\alpha)$ must therefore be a power of $P$.