Let $R$ be a commutative unitary ring that need not be local.
Let $N$ be a nonzero $R$-module.
Suppose there exists a nonzero element $x$ of $N$ and a maximal ideal $\mathfrak{m}$ of $R$ such that $x$ is killed by a power of $\mathfrak{m}$.
Let $I$ be ann$(x)$.
Then $Rx$ is isomorphic to $R/I$.
By the assumption, there exists an integer $n > 0$ such that $\mathfrak{m}^n \subset I$.
We can assume that $n$ is the least such integer.
There exists $a \in \mathfrak{m}^{n-1} - I$.
Since $\mathfrak{m}a \subset \mathfrak{m}^n \subset I$, ann($a$ mod $I) = \mathfrak{m}$.
Hence $R/I$ contains a submodule isomorphic to $R/\mathfrak{m}$.
Hence $Rx$ contains a simple submodule.
Conversely suppose $N$ has a simple submodule $L$.
$L$ is of the form $Rx$, where $x$ is a nonzero element of $N$.
Let $\mathfrak{m}$ be ann($x$).
Since $L$ is isomorphic to $R/\mathfrak{m}$, $\mathfrak{m}$ is maximal.
Clearly $\mathfrak{m}x = 0$.