What does it mean when a music library says their music is "Royalty Free" ?

Royalty-Free is more of a marketing term that denotes it will not be taken down by Youtube to users / subscribers of a music library. Every music library has different deal structures, of which you have entire control which deal structures you opt into for each track.

Some deal structures are a flat fixed fee and that's it. But most companies offer royalties in the form of Mechanicals, Publishing, Writer’s Share, Content ID, Subscription Rev Share, A La Carte Rev Share, Stream Rev Share, among many others.