A more thorough explanation:

Courts of the franchise refer to privately held courts that exist by virtue of a royal grant. These courts have jurisdiction over a variety of matters, depending on the grant and whatever powers the court acquires over time. In 1274, Edward I abolished many of these feudal courts by forcing the nobility to demonstrate by what authority (quo warranto) they held court. If a lord could not produce a charter reflecting the franchise, the court was abolished.

Franchise courts were private courts held by feudal lords. Sometimes their claim to jurisdiction was based on old pre-Conquest grants. But many of them were, in reality, only wrongful usurpations of private jurisdiction by powerful lords. These were put down after the famous Quo Warranto enquiry in the reign of Edward I.

For instance, a lord who had a franchise court could dispense justice and earn revenue from the fees and dues, fines, and amercements. However, some lords did not have a legitimate claim to jurisdiction and were only using their power to hold private courts. Edward I abolished these courts by forcing the lords to prove their authority to hold court.