In Quantum Physics it is possible for two elementary particles of opposite charges to spontaneously form. However these particles normally annihilate one another.

If these two particles were on the edge of a black hole the negative particle could be drawn into the black hole. The positive particle would escape and would appear as radiation from the b/h. As well, after an infinite amount of time, the -ve particles would annihilate the matter in the black hole and cause it to explode.

Another viewpoint of this is that the antiparticle created can be viewed as the normal particle, but traveling backward in time. In this view, the particle escapes from the black hole, traveling backward in time, and then scatters, and reverting to normal timeflow.

Yet another view is that a particle can quantum tunnel out of a black hole. The chances of this happening increase as the black hole shrinks (shorter distance to tunnel) and thus a small black hole emits more radiation than a large one. The energy output increases greatly as the black hole gets miniscule, ending in a explosion of sorts as the hole evaporates completely, having lost all of its mass/energy in radiation.

The hawking radiation corresponds exactly to perfectly normal thermal radiation from a blackbody object, and can be used to calculate the 'surface temperature' of the black hole.
As far as I know, Hawking Radiation has been attributed to Vacuum Fluctuation (read Despot's writeup) in which 2 particles are created spontaneously when quantum fluctations allow. These particles are of opposite charge and normally are attracted back to each other and the annihilate themselves.
When this happens at the event horizon of a black hole (or further out, depending on the velocity of the particles), 1 particle will be pulled into the black hole and the other will escape. Since the energy (and therefore the mass) to create these particles comes from the black hole and only half of it returns, the black hole itself has lost mass.

In reality, the mass of other outside particles pulled into the black hole is so great that most people wouldn't even notice the difference that Hawking Radiation makes on normal black holes. It only becomes apparent when it is smaller and therefore swallows less mass.
See What if a black hole was created on earth?

One very interesting fact about Hawking radiation is that it leads to the evaporation of a black hole of mass M in a time proportional to M3. Since the Schwartzschild radius R is proportional to M, the life expectancy of a black hole can also be expressed as being proportional to R3, or, proportional to the volume of the black hole. (All of this assuming, of course, that the black hole doesn't eat anything more in the process of evaporating.)

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