WordPress can send pingbacks to sites you link to. While self-pings can help connect related articles on your own site, you may not want WordPress notifying external sites whenever you publish a link. Only Self Pings restricts outgoing pingbacks to your own site or to explicitly approved URLs.
With no saved settings, it allows your current public site URL and URLs beneath it. You can replace this list with the URLs you want to allow under Settings > Only Self Pings.
The plugin does not enable pingbacks. Outgoing notifications must be enabled in WordPress, and the receiving article must accept pingbacks.