The FanPage group feature means that a FanPage can invite other FanPages to be part of their group (i.e. linking FanPages together hierarchically). This is useful with e.g. Sports Clubs, where the Club's FanPage can link all of their players "under the wings" of the club.
It's possible to create multiple levels of hierarchies, e.g. League - Sports Club - Players.
How do I activate the group feature?
You might already have it. When you created your FanPage and selected "We are a Group", then you have it. If you are not sure, you can check it at your Account Management.
In the picture above, the FanPage is an "Individual", thus not a Group. By clicking the "Show details" button, you will find a way to activate the group capability:
How do I invite another FanPage to be part of my group?