Reduction to Max Flow

You can solve bipartite matching using max flow. Create a source $s$ connected to all vertices in $L$ with capacity $1$. Connect all vertices in $R$ to a sink $t$ with capacity $1$. Use the original edges with capacity $1$.

The max flow from $s$ to $t$ equals the maximum matching size. Each unit of flow corresponds to one matched pair. This works because capacities force each vertex to participate in at most one edge.