That's a partial truth about delta wings. Even the F-16 has a delta wing. It just has a conventional tail too. The benefit of a delta wing planform, whether a tailless delta or a winged delta (or canard delta) is that it has a very flat lift curve. Yes, drag increases dramatically as AOA increases, but you also get neat vortices that keep producing lots of lift even at or after lift starts decreasing as AOA continues to increase. With a conventional wing, lift drops off rapidly resulting in an abrupt stall. With any delta wing planform, the lift tapers off gradually while drag increases at a very rapid rate. So you can fly a delta wing aircraft at a nice high AOA and still have it be very flyable since it is still controllable in the "stall".
This is one reason why the mig-21 was and still is considered a dangerous dogfight opponent. Its low speed high AOA handling was very good due to the delta wing.
A tailless delta wing aircraft like the B-58 hustler and mirage fighters rely more on the vortices than the F-16. Also, pretty much any delta wing aircraft gets HUGE benefits from a drooping leading edge flap as AOA increases.
Yea, that F-16 pilot said part of the truth. The rest of the truth is that he's flying a delta wing fighter and his flight control computers keep him from getting deep into the part of the AOA regime where drag increases super quickly as lift slowly drops off.