How are matchings a lower bound for an approximate vertex cover?
Let $M$ be a maximal matching, that is, $M$ is a matching which is not contained in a larger matching. Note that $M$ need not be a maximum matching, which is a matching of maximal size.
If $M$ contains $m$ edges, then:

Any vertex cover contains at least $m$ vertices. This is because any vertex covers at most one edge of the matching.

There exists a vertex cover containing $2m$ vertices, namely, all vertices of the matching.  Indeed, if the vertices of the matching $M$ do not form a vertex cover, then there is some edge $e$ which is not covered by the vertices of $M$, and so $M$ is contained in the larger matching $M \cup \{e\}$, contradicting the definition of $M$.