What is Hawking radiation and how does it cause a black hole to evaporate?

To add to Rory's answer-

The ability to radiate particles in a random, statistical way, is in a deep sense identical to an object having the property we know as "temperature." So, black holes have a temperature. It has a particular formula that is inversely proportional to the mass of the black hole. If you set that temperature equal to the current temperature of the Cosmic Microwave Background (CMB) that is 2.725 K, then you get a mass of about 4.503 X 10^22 kg, or a little over half the mass of the Moon. Black holes above this mass will be cooler than the CMB incident upon them, so will gather mass-energy from it. Black holes below it will lose energy due to Hawking radiation faster than they gain it from the CMB, so will head towards a catastrophic, runaway "pop." Note that the CMB is also getting cooler as time goes on, so the equilibrium mass shifts upwards. No one that I know of has bothered to do any detailed "race" calculations between a black hole's Hawking radiation and the changing temperature of the CMB.

Another important mass related to Hawking radiation is the mass at which the black hole is so cool that it would have emitted negligible radiation even if had been around since the beginning of the universe. This is about 2 X 10^11 kg, roughly comparable to the total mass of all humans.

The second mass is less than the first, so if a whole range of black holes had been created at the beginning of the universe, the upshot is that some would be popping right now! Astronomers are on the lookout for these events.

You sort of have the answer in your question - but you are assuming mass is positive, as opposed to viewing it as an amount of energy.

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.

Smaller primordial black holes can 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.

The virtual particle/antiparticle explanation is common, but (from what I understand) not very accurate; see e.g. this explanation by John Baez. To summarize it in less technical terms, spacetime near the black hole's event horizon is so strongly curved that what a nearby observer would call "absolute zero" (i.e. zero emission of radiation) looks like a greater-than-zero temperature to someone far away. That means the black hole is emitting energy, and by conservation of mass/energy the hole must be getting smaller as a result.