Non-clickbait title: "How Dropbox uses the root access that you give it during installation to give itself Accessibility authorization without triggering the usual popup".
Corrected proposed non-clickbait title: "How Dropbox fakes an authorization prompt to trick you into entering credentials that it then caches in order to bypass restrictions on what root is able to do so that it can persist a security bypass mechanism."
toomim is correct. Upon reading the update, it looks like it pops an OS X auth dialog to update a file (TCC.db) which is used to bypass the normal restrictions on what the root user is able to do. This bypass is used to manipulate the AX config.