 Solution

A many-one reduction $A{\le }_{m}B$ is a total computable function $f:\mathbb{N}\to \mathbb{N}$ with $x\in A⟺f(x)\in B$. If $B$ is recursively enumerable, compute $f(x)$ and run a semidecision procedure for $B$ on that value. It halts exactly when $x\in A$, so $A$ is recursively enumerable. Totality of $f$ is essential to this argument.