This is more like eli15, but here goes.
Empty space isn’t really empty. Much like the air around you isn’t empty, there are lots of particles bouncing around, empty space has lots of stuff too. That stuff is weird, though. Pairs of particles literally appearing out of nothing, but only temporarily, before they delete themselves again. A little debt of energy that is cancelled before it can even be noticed.
Near black holes, weird things happen. On the edge of a black hole, near the event horizon, these pairs of particles can pop into existence. But sometimes, one of the particles ends up falling into the black hole before it gets to disappear with its counterpart. That counterpart can escape away from the back hole. Because stuff can’t actually come from nowhere (conservation of energy), the black hole must lose a little mass in order for this particle to leave the black hole. This is a teeny tiny amount of loss, and since it happens only at the event horizon, the amount of loss is related to the surface area of the black hole, proportional to its radius. The total energy in the black hole is proportional to the cube of its radius, and black holes are very very big. So it takes a long, long time for hawking radiation to do anything.
As the black hole shrinks, the surface area becomes bigger relative to the volume, and the rate of loss speeds up. By the end of the life of a black hole, it’s shrinking very fast, and all of the remaining energy ends up getting released quite quickly; assuming that hawking radiation works the way we think it does, the final evaporation of a black hole is quite a big burst of gamma rays, which you don’t want to be anywhere near.
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