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How does adding a particle to the black hole remove energy from it?
The entire reason the particles can come in to existance is because the black hole curves spacetime enough to 'eat' one of the pair. It only exists because of the black hole. The particle leaving the black hole takes energy away because that area of spacetime now has less energy in it, meaning the black hole shrinks. The black hole isn't magically adding energy to the space around it in order to create these pairs.
If you throw a ball away from you, yes you feel the force, but now you've sent a bunch of energy away from yourself in the ball. In effect, the black hole is 'throwing' particles away from itself by the simple act of eating part of the spawned particles.
I've explained it poorly, but PBS Spacetime has several great episodes on the specific phenominon.
It's not a particle in the regular sense you might know, like an electron. The pair that comes into existence is meant to annihilate immediately (meaning there's zero energy gain or loss) but because of where it appeared it can't.
When it appears as I described, there suddenly exists a real particle in the universe outside the black hole, so the universe gained +1 in energy.
But energy can't be created or destroyed, so that +1 means somewhere there must be a -1. And that somewhere is the black hole which caused the particle to exist in the first place by swallowing its pair.
It's not very intuitive, that's the fun part about quantum mechanics: nothing is intuitive.