Proving flight are a practical demonstration of the table tops, basically from the theory (table tops) to "Prove" that the theory is correct. Usually this is a smooth seamless event, however sometimes it goes in the wrong direction, you need a very experienced crew when you do this.
FAA will have several specialists on board. Flight Operations Inspector to observe the flight procedures, Cabin specialist for cabin crew, Maintenance specialist for the mechanical problems that FAA will artificially create, Nav Specialist etc.