AGPL is based on modification rather than public performance or use. You don't have permission to modify copyrighted things by default so if you do modify, then you either comply with the AGPL or violate copyright law.
Because a number of court cases have sided with the interpretation that "copying bits into RAM for execution" is itself governed by copyright law, in practice you wouldn't be able to use your modified software without breaking copyright. This is why software licenses are "needed" -- lacking some kind of limited license to make copies, execution of software isn't permitted.
