It adds a suid binary to /Library/DropboxHelperTools. This binary executes as root's effective user ID no matter who executes it and adds Dropbox to the accessibility list.

Dropbox doesn't save your password.

...But I am wondering why one of these suid binaries is world-writable.

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.

Slimy, slimy, slimy.