Two sides and the pairing goal
A bipartite graph splits nodes into two groups with edges only between the groups, never inside one. Think of workers on the left, jobs on the right, and an edge wherever a worker can do a job. A matching is a set of edges where no node is used twice. The maximum matching pairs up as many as possible.
Augmenting paths again
A path that alternates between unmatched and matched edges, starting and ending at unmatched nodes, is an augmenting path. Flipping which edges along it are matched increases the matching size by exactly one. Keep finding augmenting paths until none remains; the result is a maximum matching.
Reduction to flow
You can also solve it as a flow problem. Add a source linked to every left node and a sink linked from every right node, give each edge capacity one, and the maximum flow equals the maximum matching. Each unit of flow is one chosen pair.
Konig insight
In bipartite graphs the size of a maximum matching equals the size of a minimum vertex cover, a small set of nodes touching every edge. This pleasing duality mirrors max flow min cut.
Key idea
Maximum bipartite matching pairs two groups using augmenting paths, and it reduces cleanly to a unit capacity max flow problem.