In the flow network, integer max flow sends at most 1 unit through each vertex (by capacity constraints). A flow of f units means f edge-disjoint paths from s to t, each passing through one vertex in L and one in R.
This set of paths corresponds exactly to a matching of size f. Since max flow finds the largest such flow, it finds the maximum matching. Algorithms like Dinic or Ford-Fulkerson solve this in O(V2E) or faster.
Space complexity is O(V) for the data structures used.