I think I know what they mean. They're talking about a false vacuum scenario.
A vacuum state is a state of lowest energy. It's thought that the vacuum of our universe is in a lowest-energy state and is stable, and so nothing special will happen to spacetime. However, if our universe is actually a false vacuum, then it could merely be metastable, and some tiny perturbation could cause it to fall into a lower energy state - a true vacuum, or else a false vacuum of lower energy. The "bubble" that would appear would be a region of this lower-energy vacuum that would expand across the universe without stopping.
Scientists aren't sure if our universe is a false vacuum. Here's a chart of the masses of the Higgs boson and the top quark:
The latest measurements suggest that their masses lie in a metastable region of the graph (which could be bad), although it's towards the stable end of the region (which is good). This vacuum catastrophe could still happen, but the odds aren't in its favor. I also suggest reading some of the excellent papers in the references section of the Wikipedia article. They're quite comprehensive.
The universe can be thought of as space containing a variety of quantum fields. A quantum field might be best thought of as something that has a value at every point in space. Put together a bunch of these fields and you can describe particles, particle interaction, and all the matter and energy in a given region of space - in fact, in the whole universe!
Picture a region of space with absolutely nothing in it. Nothing. (I'm ignoring vacuum energy, even though I really should discuss it) That's a true vacuum. Remove all the fields, particles, and other interesting stuff from our universe and that's what you'll get. Nothing.
The false vacuum scenario happens if there isn't actually "nothing" but "something". If our "vacuum" has some extra energy that it shouldn't (again, I'm not talking about normal vacuum energy). In other words, there's something where there should be nothing.
In this end-of-the-universe scenario, the universe goes from having this something to having nothing - from a false vacuum to a true vacuum. The region of space actually having nothing starts at a certain region and expands outward at the speed of light. That's what the scientists were talking about.
For anyone who wants a further explanation and/or could tell how many mistakes I made there: Yes, I'm aware that there were mistakes in that incomplete explanation. I didn't properly explain vacuum energy or an energy state, nor did I discuss stability or even properly touch on quantum fields. But quite frankly, I don't think this explanation needs that baggage. Does it add to the richness of the concept? Yes. But is it confusing? Also yes. I don't think working that in would be productive.