We start with the following
Proposition 1. A graph $G$ is $k$-edge colourable if and only if the edge set of $G$ can be decomposed into (at most) $k$ matchings.
The above would follow immediately once you recognize that each color class of edges corresponds to a matching. With this new terminology at hand, we can rewrite the given statement equivalently as
Theorem 2 (Hall's theorem for bipartite regular graphs). Suppose $G$ is a bipartite $k$-regular graph. Then the edges of $G$ can be decomposed into $k$ (perfect) matchings.
This is a standard "big" theorem in graph theory. So any justification of the OP's statement must either quote Hall's theorem directly, or must reinvent the wheel and prove Hall's theorem implicitly. I prefer the first approach over the second. See the wikipedia page for more details on Hall's theorem.
Note: Often, textbooks prove the seemingly weaker statement:
Theorem 3 (Hall's theorem for bipartite regular graphs, version II). Every bipartite regular graph contain a perfect matching.
But the two versions are, in fact, equivalent. To go from Theorem 3 to Theorem 2, keep stripping off a perfect matching from the graph repeatedly $k$ times, until the graph becomes empty.