“Edit this entry” is a WordPress administration tool. It was designed to make blogs and sites easy to update. Clicking the link will pull up the post/page you are viewing into the Admin editor box.

The “edit this entry” link only appears when you are logged into the admin. You can remain logged in for up to two days unless you specifically select the “Log Out” link in the top right of the Admin dashboard.

Rest assured this link is not visible via other computers.