Why is bipartite perfect matching a special case of clique problem?
Let $G=(V,E)$ be the bipartite graph.  Construct a new graph $G'=(V',E')$ whose vertices are the edges of $G$, and where we add $(e,e')$ to $E'$ if $e,e'$ are two edges from $G$ that don't touch each other (don't have an endpoint in common).  Now there's a matching of size $s$ in $G$ if and only if there's a clique of size $s$ in $G'$.  In particular, there's a perfect bipartite matching for $G$ if and only if there is a clique of size $|V|/2$ in $G'$.