This is a new (v4.x) function accessible via the Sidebar (options 3-2) and lists all standard missions available at the time. When the Skunk enters a new zone or sector all the available mission will be added to the BBS. Exceptions are the hidden or secret missions: in order to add them to the BBS they must first be found by flying close to the surface of stations (same as prior to v4.x). Missions will stay in the BBS for a limited amount of time or until you go out of range such as when travelling through a Jump Gate.
Please note that if you're loading an existing save after updating to version Rebirth 4.0, you'll need to change zones before the BBS will begin to populate.