Solving the min edge cover using the maximum matching algorithm
Assume, w.l.o.g., that G=(V,E) is a graph without isolated vertex. Let's denote by $\mathcal{A}$ the algorithm you describe in your question. We seek to prove that given $G$, $\mathcal{A}$ outputs a minimum edge cover. Before we do that, it can be benifical to make some simple observations. First, both edge cover and matching are set of edges, and the subset of an edge cover can be a matching. Given a minimal edge cover $C$(not necessarily the minimum), we denote by $M_C \subseteq C$ the maximum matching containing in $C$ . Note that $M_C$ is the matching of maximum size whose edges are all in $C$. We call any edge in $M_C$ the matched edge. We use $V(M_C)$ to denote the vertices of edges in $M_C$
Observation 1 Every vertex in $V\setminus V(M_C)$ must be covered by an edge which is incident to an edge in $M_{C}$. Otherwise we could find a larger matching in $C$.
Observation 2 Every edge in $C$ covers 1 or 2 vertices. Specifically, an edge in $M_C$ covers two vertices, an edge in $C\setminus M_C$ covers 1 vertex.
Theorem 1 $|V|=|C|+|M_C|$
Proof $\;$  By observation 2 and the definition of edge cover, we have $2|M_C|+|C-M_C|=|V|$.
By observation 1 we have that algorithm $\mathcal{A}$ outputs a minimal edge cover. Then he correctness of algorithm $\mathcal{A}$ is obvious from theorem 1:  when $|M_C|$ is maximized, $|C|$ is minimized since they sum to a fixed number $|V|$. In such situation, theorem 1 is called Gallai’s theorem