Firstly, I'd like to apologize if I've posted this in the wrong section, as I wasn't sure if it belonged in Relativity, Quantum Theory, Modern/Theoretical Physics, or here in Astronomy and Cosmology; my question really relates to all these topics in one way or another.
Anyway, on to the meat of my query:
As I understand, "Hawking Radiation" is a result of the meeting between Heisenberg's Uncertainty Principle and the incredible gravity at and around the event horizon of a black hole; as a particle-antiparticle pair "pops" into existence, one of these particles is sucked into the hole and the other, in turn, is cannoned off into space, (thus the radiation) rather than their usual collision and evaporation. What I don't understand about this proposed phenomenon is how it is responsible for the evaporation of the Black Holes, seeing as the radiation isn't being emitted from the Black Hole itself, but rather from the quantum fluctuations of spacetime around it. It seems to me that it would cause only the accretion of more mass—being that the Black Hole is swallowing particles that "pop" into existence around it—rather than the evaporation of it.
Can someone explain to me what it is about this process that results in the evaporation of the black hole?