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.
