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Very often a virtual particle–antiparticle pair appears and because they're very happy they started existing, they immediately hug each other not knowing that will cause them to annihilate each other and disappear.
Every once in a while, the pair appears in a very interesting position: one is outside the event horizon of a black hole (let's call this one Pinocchio) and the other inside it (let's call it 3735928559). Because nothing can escape a black hole, they can't really hug, so Pinocchio says "I'm a real ~~boy~~ particle" and stops being virtual and becomes real, while 3735928559 continues its descent into ~~madness~~ singularity.
Unfortunately, the process means there now exists something (Pinocchio) where there wasn't anything before and that takes energy. And that energy comes from the particle that stayed behind which is now part of the black hole, so it effectively takes energy out of the black hole. You may have heard that energy cannot be created or destroyed, only transformed, that's essentially what happens here.
The Pinocchios go away from the black hole, so they can end up basically anywhere in the universe.
As for the timescales, they entirely depend on the black hole's size. Really tiny black holes evaporate in a matter of seconds, the supermassive ones in a matter of trillions of trillions of trillions of trillions... years.
In fact, black holes will be the last macroscopic structures to exist in the universe because the evaporation is extremely slow - every planet and every star and every gas cloud and every atom will cease to exist long before the last black hole evaporates.
How can stuff have negative mass in the true sense of that?
In this context, "negative mass" is a mathematical convenience rather than an actual particle having negative mass.
Think of it more like "energy required to pull apart a matter-antimatter pair". In the vacuum of space, the energy that created the pair gets returned when they annihilate. But when near a blackhole, it had to "burn" some of its energy to interrupt that process. Energy is mass, so the blackhole gets less massive.
Mind you this is a very basic explanation of it. It's just another quantum whackyness of our universe.