Arkansas

They hinge on a few things.

First, the first amendment doesn't cover the irredeemably obscene , incitement to imminent lawless action, or things like slander and libel. Free speech isn't absolute.

Second, minors don't have the same first amendment rights as adults. There's a general agreement that there's some theoretical manner of content that isn't suitable for minors because they can't contextualize it correctly.

Third, despite the real and increasing threat to freedom of speech in the US, we still have enough protections that what we (correctly) angrily call "book bans" are not what the phrase conjurs in isolation.
It's typically the government refusing to endorse or provide the book for biased or political reasons. Bad, but not the government prohibition of speech that the phrase evokes.

Basically, they gain any traction at all because they're not banning a book, but asserting that the content of a book is too obscene to give to minors, and then trying to criminalize that.

It's preposterous in both principle and application, and particularly monstrous given how they're being used to target the most vulnerable people, but they're based off the legal grey area that's supposed to be filled in by well intentioned reasonable people because the law as written can't account for every possibility.