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In depth-first search of an undirected graph, every edge joins a vertex to one of its ancestors or descendants in the discovery tree. To see this directly, suppose $u$ is discovered before its neighbour $v$. If $v$ is still undiscovered, the search from $u$ cannot finish before discovering $v$: it eventually scans $uv$. Thus $v$ is discovered while the call for $u$ remains active and belongs to its descendant subtree.^[ref]

For any edge $uv$, its ancestor endpoint therefore has a proper descendant and hence at least one child. It is an internal vertex of a rooted tree. Every edge has an internal endpoint, so the internal vertices form a vertex cover. The relevant leaves of a rooted tree here have no children in the rooted discovery tree; a root of vertex degree one can still be internal.