Reduction from a further constrained problem
Hint: An instance of densest $k$-subgraph can be viewed as an instance of your problem, in which sets are vertices and elements are edges.

More generally, to prove that a problem P is NP-hard given that a (seemingly) related problem Q is NP-hard, you need to come up with a polytime reduction from Q to P. If Q is strongly related to P, such a reduction could be easy to find.