Sometimes themes include an [Edit] link as a shortcut for logged-in users to get to the edit screen quickly. If you log out of wp-admin, this should go away. The public wouldn’t see this on the site.
Depending on your theme, you might be able to switch it off with an option, but if not, you could edit the theme to remove the link, but it doesn’t do any harm since it would typically only show up for users that have permissions to edit that post.