The answer is that the particles do not come from within the black hole, but from the “empty” space just outside the black hole’s event horizon.
This understood in the following way: what is thought as “empty” space cannot be completely empty, because this would mean that all the fields, such as the gravitational and electromagnetic fields would have to be exactly zero. However, due to the uncertainty principle (which states that the more accurately you measure the position of a particle in space, the less accurately you could measure its velocity) the field in “empty” space cannot be fixed at exactly zero, because it would then have a precise velocity (zero). Therefore there must be a certain minimum amount of uncertainty in the value of the field.
Due to scientific theory every matter particle has an opposite match; or if you like an antiparticle. If these particles are seen as matter particles, then one of the partners of the particle/antiparticle pair will have positive energy, and the other negative energy. The one with negative energy will be condemned to be a short-lived virtual particle because real particles always have positive energy in normal situations. It must therefore seek out its partner and annihilate with it.
However, a real particle close to a massive body has less energy than if it were further away, because it uses energy to lift it far away against the gravitational attraction of the body. Normally the energy of a particle is still positive, but inside a black hole gravity is so strong that even a real particle can have negative energy there. It is therefore possible for the virtual particle, with negative charge, to fall into the black hole and become a real particle. In this case it no longer has to annihilate with its partner.
Its forsaken partner may fall into the black hole as well. Or, having positive energy, it might also escape from the vicinity of the black hole as a real particle, or antiparticle. To an observer (obviously at a distance, as they would cease to exist inside a black hole) it would appear to be emitted by the black hole.
The smaller the black hole, the shorter the distance the particle with negative energy will have to go before it becomes a real particle and thus the higher the rate of emission, and the apparent temperature of the black hole.
What happens when the mass of the black hole becomes extremely small is not yet clear, but a reasonable guess would be that it would disappear finally in a tremendous burst of emission, equivalent to the explosion of millions of hydrogen bombs.