It follows from the premise. If everything is meaningless, then even meaninglessness is meaningless. Furthermore, the fact that it is meaningless doesn’t preclude us from assigning it any meaning we want since that process too is — you guessed it — meaningless.
It’s like taking a test with no wrong answers since all the answers are ultimately wrong. You’re free to answer any way you want — or not even take the test — because you can’t score any better or worse than the person who tries.
It is self-contradictory and fundamentally incoherent to invoke the universal, invariant, objective laws of logic to oppose objective knowledge. An autonomous subjectivist or relativist cannot justify the standards of rationality to which he appeals. His attempt to destroy rationality and knowledge is self-refuting.