The AuthorProtect extension allows page authors to protect pages from non-authors on a page-by-page basis. Authors with the 'authorprotect' permission will gain a protect tab on pages they author that gives them access to this interface. Because logs and anonymous users don't mix very well (as with the fact that all anonymous users share the same user id), anonymous users are unable to use this extension regardless of if they have the 'authorprotect' right or not.
Do not assign the author right via GroupPermissions (except to those who you want to be able to edit author-protected pages)! This right is automatically assigned to the author of the page. Assigning this right via $wgGroupPermissions gives that group the ability to edit all author-protected pages, regardless of if they are the author of the page or not (by default user group administrators have this right).