To expand some on DeltaLima's answer:
Pilots on an Instrument Flight Rules (IFR) flight plan are afforded more separation between them and other planes/terrain/obstructions than pilots operating under Visual Flight Rules (VFR). This extends to landing at an uncontrolled airport as well; if an aircraft retains its IFR clearance all the way to the ground, ATC will "protect" the airspace around that airport and will not let any other IFR aircraft arrive at or depart from that airport for a certain amount of time. The airport is considered "one-in one-out" (or I suppose more correctly "one-out one-in") for IFR flights. If the pilot doesn't get in touch to cancel the flight plan search-and-rescue procedures will be initiated.
In this case the controller had a Remote Communications Outlet at the satellite airport. This is an antenna and radio transceiver connected to the radar facility via telephone lines or similar, allowing the pilot to contact the controller directly when on the ground. If an RCO is not available the controller will provide a telephone number that reaches a recorded line at the control position and the pilot can cancel that way.
Most likely the controller said "Report cancellation [of IFR] in the air this frequency or on the ground with me at this other frequency." Or perhaps the pilot had already been switched from the main frequency to the RCO, and was told "Report cancellation on the ground with me." The pilot responded that they would cancel on the ground, instead of in the air before arriving. (If the pilot cancels in the air the airport is not "blocked" for other IFR traffic, which is helpful when there are multiple arrivals in sequence.)
As an aside, it is said that at controlled airports the IFR flight plan is closed "automatically." I had a little trouble understanding what this means when I was first learning about it. It isn't an automated function by the computer that flags the flight plan closed instead of open; rather means that, by default, no one will come looking for you or ask you to report cancelling. Instead the tower controller observes you landing safely, and if you don't will initiate rescue operations.