those aren't cosets, just subsets of G made from subgroups.
on the LHS, you have elements of the form hk, h in H, k in K. this is also an element of N (because of the intersection sign).
so we have hk = n. but h is in N, since H < N, so h^-1 is in N (N is a subgroup), so k = h^-1n is in N.
thus the k in the hk is in K and N, that is, it is an element of K∩N. this shows HK ∩ N is contained in H(K ∩ N).
the containment the other way is even easier to show. have fun.