Why is one traversal sufficient for the Kuhn's maximal matching problem algorithm?
Kuhn's algorithm maintains the following invariant: after scanning through vertices $v_1,\ldots,v_k$ on the left, the current matching is a maximal matching of the graph consisting of $v_1,\ldots,v_k$ on the left, and the entire right-hand side. Hence at the end, we get a maximal matching of the entire graph.

How do we know that Kuhn's algorithm maintains the invariant? We prove it when we prove that Kuhn's algorithm is correct. I encourage you to find a correctness proof of the algorithm (such proofs are surely not too hard to find online) and read it.