In this article, he describes what Hawking's last paper is all about, if you want simple description of it. The link to the preprint (we'll update this post if and when it is published) is also given if you don't have it already.
Here is, in a nutshell, what they do. They create a (deformed) conformal field theory that is mathematically equivalent (or dual) to an eternally inflating spacetime, and investigate some mathematical properties of that field theory. They look, in particular, at where the border of a spacetime that inflates for an eternity (forward in time) versus one that doesn't, and choose that as the interesting problem to consider. They then look at the geometries that arise from this field theory, try to map that back onto our physically inflating Universe, and draw a conclusion from that. Based on what they find, they contend that the exit from inflation doesn't give you something eternally inflating into the future, with disconnected pockets where hot Big Bangs occur, but rather that the exit is finite and smooth. In other words, it gives you a single Universe, not a series of disconnected Universes embedded in a larger multiverse.
There! Do you even need to read the actual paper after that?
BTW, let's also give some love to his co-author, Thomas Hertog, who seems to be left out in many of this discussion and news articles.