How to prove fooling set problem to be NP-hard
The proof is given in Section 6 the full version of the paper. I bring an excerpt, since there was originally much confusion about what "fooling set" means.

Basically, the notion of fooling set was introduced by Jean-Camille Birget in Intersection and union of regular languages and state complexity.

To prove that fooling set problem (as defined below) is NP-hard, the authors reduced the NP-complete induced matching problem on bipartite graphs to it.

Here's an excerpt of the paper, which defines the problem and proves the theorem.