As a lay summary, they show that if there was an equivalent to the 2nd law of thermodynamics for entanglement, then you could use 2 copies of states to create 3 copies of states. However, that would violate the no copy principle, and other mathematical axioms. Therefore, there is no 2nd law of entanglement.

Note, they also show that you could get a 2nd law if there exists a macroscopic amount of new entanglement being fed to the system.