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
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.