Users logged-in as admins can control links from the front-end, starting from version 1.2.6 of the WordPress Curated Lists plugin.
The following animation highlights the three icons only admin see:
Editing links on the front-end
Editing links on the front-end
Opens the category settings
Opens the settings of this link
Deletes the link (after a confirmation prompt)
Settings
To display admin links on the front-end, head to the Plugin Options tab, enable the option Edit/delete link for admin in frontend, then save the settings. Note: this option was introduced in version 1.5.9.