 Solution

Choose a maximum matching $M$ and a minimum vertex cover $C$. By Kőnig's theorem for bipartite matching, $|M|=|C|$. Every edge of $M$ must have at least one endpoint in $C$, because $C$ is a vertex cover. Distinct edges of the matching in a graph have disjoint endpoints, so these requirements already use at least $|M|$ distinct vertices of $C$.

Since $C$ has exactly that many vertices, each edge of $M$ has exactly one endpoint in $C$, and no vertex of $C$ lies outside the endpoints of $M$. Thus selecting the unique endpoint in $C$ from each edge of $M$ produces precisely the vertex cover $C$:The argument also covers an edgeless graph: take both sets empty.