Does anyone know what this functionality is called and how it can be implemented? I do not have any of these apps added to my Facebook account but every so often I get what's shown in the screenshot in my newsfeed. This is a great way to get click-through's into an application.
A lot of content-delivery sites have this functionality e.g. news sites, and even Pinterest now. See:
The Apps and Games "app" is made by Facebook and has access to functionality not (yet) available to regular 3rd party developers.
The closest thing we can do is aggregation reports: http://developers.facebook.com/docs/opengraph/define-units/
The Pintrest report looks more like what can be done by us mere mortals.
See for example this report from an app that I made:
(source: jocke.syk.se)