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A plate is something that exists within all living beings; it holds the Quirk of every person. It's not something found in the body or the soul, it's something between.
Each time a Quirk it's used, it's like a drum: the larger it is, the greater the sound made or the power used, that depends on the soul; the thicker it is, the longer the sound lasts or the more it can be used without consequences, that depends on the body.
What AfO's does when he takes Quirks is compact them and store them in his plate in such a way that he can't use them at their full potential, but he can use them all. Power through versatility.
By sharing them, he causes another person's plate to receive them, if it's too small, they lose their sanity and become an irrational being; if the plate is too thin, it breaks, and the person dies.
That's how the Nomus come about.
…Oh, and your sensei?
He has the biggest plate I've ever seen, probably the biggest in all of humanity. But it's also fragile. I think the only thing that's kept it from breaking is that it hasn't truly made them its own.
Let me explain. Quirks carry with them an echo of their owners' will, something like ghosts or replicas. If someone takes a Quirk that isn't theirs, they have to eliminate that echo for it to truly be theirs and for them to unleash its full potential.
The only thing keeping Cale Henituse alive is that he refuses to absorb those echoes.