 Solution

A many-one reduction $A{\le }_{m}B$ is a total computable function $f$ such thatIf $B$ is a computably enumerable set, compute $f(n)$ and run a semidecision procedure for $B$ on that value. It halts exactly when $f(n)\in B$, equivalently exactly when $n\in A$. Therefore $A$ is computably enumerable.