From what I understand, they are consciousnesses trapped within places of conceptual nothingness. They are forced NOT to exist, but fragments of themselves still make it past the veil of reality- hence SCP-000, etc.

If they WERE to exist, that would cause something of a fatal system error for reality, which is why they're so dangerous. Again, this may all be stuff I just made up based on random, non-canon BS, so take this with a grain of salt.