The graph induced by a matching has maximum degree 1, and a graph whose edges are formed from the union of two matchings $M_1,M_2$ has maximum degree at most 2. Thus, we have a graph $G'=(V,M_1 \cup M_2)$ whose maximum degree is at most 2. A graph having maximum degree at most 2 is necessarily a disjoint union of paths and cycles (for if a component contains a cycle, it can't contain anything else besides this cycle since a vertex cannot have degree more than 2). To prove $G'$ is bipartite, we need to prove that if one of the components of $G'$ is an odd cycle, there is a contradiction. Assign each edge in the first matching color 1, and each edge in the second matching color 2. Thus $G'$ can be properly edge-colored using at most 2 colors, whereas an odd cycle requires 3 colors for its edges.