Normally, a black hole is considered to draw all matter and energy in the surrounding region into it, as a result of the intense gravitational fields. However, in 1972 the Israeli physicist Jacob Bekenstein suggested that black holes should have a well-defined entropy, and initiated the development of black hole thermodynamics, including the emission of energy.
In 1974, British physicist Stephen Hawking worked out the exact theoretical model for how a black hole could emit black body radiation.
In a simplified version of the explanation, Hawking predicted that energy fluctuations from the vacuum causes the generation of particle-antiparticle pairs of virtual particles near the event horizon of the black hole. One of the particles falls into the black hole while the other escapes, before they have an opportunity to annihilate each other. The net result is that, to someone viewing the black hole, it would appear that a particle had been emitted.
Since the particle that is emitted has positive energy, the particle that gets absorbed by the black hole has a negative energy relative to the outside universe. This results in the black hole losing energy, and thus mass (because E = mc2).
Smaller primordial black holes can actually emit more energy than they absorb, which results in them losing net mass. Larger black holes, such as those that are one solar mass, absorb more cosmic radiation than they emit through Hawking radiation.
Hawking radiation was one of the first theoretical predictions which provided insight into how gravity can relate to other forms of energy, which is a necessary part of any theory of quantum gravity.
Though Hawking radiation is generally accepted by the scientific community, there is still some controversy associated with it. There are some concerns that it ultimately results in information being lost, which makes physicists uncomfortable. Alternately, those who don't actually believe that black holes themselves exist are similarly reluctant to accept that they absorb particles.