Not having the necessary background in physics, I'm not quite sure what this means. I'm sure some pop-science site will break it down later, but is the gist that this has implications on our understanding of the way that black holes expand?
I don't think so. Theory of black hole formation predicts they will spin (although emitting energy slows the spin). They have different predictions for the kind of spin or lack thereof which would produce different observations -- and the data gathered supports one spinning model over another, either of which agrees with theory. Non spinning black holes would be perplexing.
What does it mean for a black hole to spin? Does that mean that matter past the event horizon is in motion, and information about that is somehow being leaked out? Or just that matter outside the event horizon is circling the drain?
Matter past the event horizon falls into the singularity pretty quickly, and that's a mere dimensionless point in space. But the singularity's spin does have an effect on spacetime around the black hole, in effect creating a second (much stranger) type of event horizon called an ergosphere. In a way one could say that rotation changes the shape of the black hole and that property can be observed from the outside to determine the spin. This is cool because, while pretty much everybody was certain that black holes do spin, this hypothesis was not yet backed up by data. The phenomenon doesn't have anything to do with quantum information leaking out, however.
I didn't get to see the actual announcement but my guess is that effect of the spin on spacetime has now been actually observed in the wild.
Udo is correct, but I want to clarify: the natural result of rotating matter undergoing gravitational collapse is a rotating black hole, and this "spins" in a way which is independent of any matter remaining outside of its event horizon. The spacetime itself has a well-defined notion of angular momentum, and there need not be any matter remaining at all.
The mathematical model for this in the language of general relativity is the Kerr metric, named after its discoverer Roy Kerr:
In many ways, a spinning black hole is much like an ordinary mass spinning at the same rate. If frame dragging (a prediction of general relativity) turns out to be observable, we should see black holes affecting the masses around them as a result of the spins of (a) the black hole and (b) the affected masses.
Only if none were ever observed to be spinning. One non-spinning black hole is like one planet with a perfectly circular orbit -- we can chalk it up to chance.
The reason black holes possess spin is because spin angular momentum is conserved, and the masses that fall into the BH each have spin angular momentum. The black hole's spin angular momentum is therefore the sum (so to speak) of the constituent masses that make it up.
http://www.nasa.gov/mission_pages/nustar/news/telecon2013022...
Spoiler: it's the first measurement of a black hole spin.