Hawking radiation escapes the horizon through quantum tunneling. Hawking calculated the radiation escaping through a quantum field theory in a general relativity classical background. It can also be understood as virtual particles escaping from outside the horizon and falling in with negative nergy, equivalent to positive energy particles escaping. These quantum effect causes the black holes to loose mass and energy in the form of different kinds of radiation escaping.
Note that it does not violate the theorems of black holes because those do not apply to quantum effects.
The calculations he and many other have done were done in the gravitational field near the horizon, which can be represented as a classical general relativity metric, with the matter being a quantum field. Closer to the singularity it does not apply - as you get closer to the sigularity the classical general relativity metric becomes less and less valid, as quantum gravity effects enter in and we do not know how to represent or calculate those. General Relativity becomes invalid as you approach the singularity. It has nothing to say about what happens there or nearby.
The above is now commonly accepted in black hole physics (though possibly I didn't state it all perfectly. The next paragraph is a little surmising on my part, at least on the conclusion, and may or may not be commonly accepted. Either way, what happens at the singularity, or anywhere near where quantum gravity could enter in, is just unknown at this time.
Now, if the Hawking black body radiation theorem is right, it says the black hole radiates at a temperature determined by his effect, and nothing else enters in. Since whatever happens at the singularity, if it is causal (a real singularity would not be, but then there would be no physics for it), it could only affect the energy coming out of the horizon through interactions with the classical gravitational field near it, it seems that anything else that happens inside the horizon might be accounted through the Hawking blackbody radiation already calculated.
EDIT after comment below: the explanation in the paragraph before the last that the 'above is commonly accepted in black hole physics' is just that. It's the theory of black holes, some of which has been observed. But Hawking radiation has not yet been observed, it may be difficult to observe it for macroscopic black holes but it is not impossible that astrophysical observations of the density of black holes and their masses over the time of the universe's evolution could confirm that they'd have to evaporate at the Hawking rates. Again, nothing measured at th time.
