How does the Hawking Radiation mechanism cause a black hole to lose its mass? Correct me if I am wrong: in the Hawking Radiation mechanism, when a virtual particle-antiparticle pair gets created at the edge of the black hole, a black hole could sometimes eat up one of the particles before it could annihilate with the other particle, and this causes the other particle to radiate away, and this process causes a loss in the mass of the black hole.
What I don't understand is: if a black-hole also absorbs a particle from the pair, shouldn't the black hole actually be gaining mass? Since the black-hole hasn't lost anything but instead gains a particle?
 A: The black hole initially lost the gravitational energy that was needed to create the pair. The pair-creation model is a bad description of Hawking radiation, which for macroscopic black holes is really photons. The second particle that gets created above the event horizon doesn't have nearly enough energy to escape. It does, however, produce photons above the event horizon, some of which can escape after being red-shifted very strongly. What we would see is therefor black body radiation escaping, but as long as black holes are much colder than the universe not even that can happen. See http://en.wikipedia.org/wiki/Hawking_radiation for the details. 
A: According to p. 303-304 of the book Gravity from the Ground up by physicist Bernard Schutz, viewable on google books here, it's because in terms of the pair-production explanation for Hawking radiation, one member of the pair actually has negative energy and thus causes the black hole to lose mass (negative mass/energy falling into a black hole can also cause it to lose mass and decrease in radius in classical general relativity, see the second paragraph of my answer here). From those pages:

Quantum theory allows uncertainties and fluctuations that are not
  allowed in non-quantum physics. Temporary fluctuations can produce
  photons of negative energy. In order to preserve the total energy,
  negative-energy photons form in pairs with positive-energy partners.
  These pairs almost immediately re-combine and disappear, since the
  quantum theory has to get rid of the negative-energy photons quickly
  in order to produce macroscopic physics of positive energy. But
  negative energy does exist for short times, in these quantum
  fluctuations.
...
How can black holes emit radiation? It should be no surprise that the
  answer lies in quantum uncertainty. All over spacetime the quantum
  electromagnetic field is undergoing the little negative-energy
  fluctuations that we considered above. Normally they are harmless and
  invisible, because the negative-energy photons disappear as quickly as
  they form. But near the horizon of a black hole, it is possible for
  such a photon to form outside the hole and cross into it.
Once inside, it is actually viable: as we remarked earlier, it is
  possible to find trajectories for photons inside the horizon that have
  negative total energy. So such a photon can just stay inside, and that
  leaves its positive-energy partner outside on its own. It has no
  choice but to continue moving outwards. It becomes one of the photons
  of the Hawking radiation.

In this answer John Rennie gives some more explanation of the mathematical derivation of Hawking radiation that this verbal description is meant to serve as shorthand for; I'm sure you need a good technical understanding of the mathematics of quantum field theory to really understand it though, verbal descriptions can only give you a flavor. 
