From the little I understand, the following is a drastic oversimplification, and is somewhat untrue, but it is a good way to visualize the process.
As you might have heard, at the quantum level, you can have particles pop out of the vacuum (a particle-anti particle pair), and then quickly combine and dissipate. This does not violate the Law of the conservation of Energy because of how fast that process happened, and in the end when they disappeared the total energy was the same as before (also an oversimplification, but take my word). Another thing you need to know is that anti-particles and particles both have positive mass and energy.
Now imagine one of these particle - anti-particle pairs appeared right on the edge of an event horizon in a black hole, now they broke free, but when they tried to recombine one of the particles was in the event horizon, and one was not. Because of this, they could not recombine and even out the total energy. But because physics says that can't happen, the way you explain the increase of total energy (represented by the particle which didn't fall into the black hole), is that the particle that did fall in had a negative energy, and when that negative energy entered the black hole it reduced the total mass of the black hole.
