Several other comments talk about a pair of particles being created out of nothing, one gets absorbed and the other flies away. This is basically right, but can be confusing (the one that gets absorbed has negative energy in order to conserve energy). Here's an easier mental model....
Steve Hawking came up with an idea a while ago (70's perhaps?). He was thinking about black holes whose event horizon was around the size of an atom. Then he put it up against the Heizenberg Uncertainty Principle. He realized that particles in these black holes would have such a high degree of certainty about their position, that there would be such a low certainty about their velocity. Therefor, there would be some that would be REALLY fast. Not fast enough that they could escape the pull of the black hole, but fast enough that they could get just above the event horizon. There, they could give off a high-energy photon, and fall back in. This photon, since it was emitted outside the event horizon, would actually escape. This radiation can (and has been) detected, and causes what is known as evaporation. http://en.wikipedia.org/wiki/Hawking_radiation#Black_hole_evaporation
Ironically, this means that smaller black holes (which have higher certainty about a particle's position) evaporate faster. Large-ish black holes absorb more energy cosmic microwave radiation than they emit in Hawking radiation, but if they have small enough mass (I believe smaller than the size of our moon), they emit more Hawking radiation than they receive from the cosmic background.