Why do extended landing flaps let a plane fly slower without falling, even though they also create extra drag?

Answer: Increase the wing's camber and effective area

Increase the wing's camber and effective area ✓ — Right. Flaps increase the wing's camber and often its effective area, so the same speed produces more lift. The drag penalty is not a bug during landing; it helps the airplane descend steeply without accelerating. The neat trade is that landing flaps make the wing both slower-flying and draggier.

Only add drag to the wing — Close to a common cockpit impression, but incomplete. Flaps do add drag, so they can feel like air brakes during landing, yet their bigger low-speed trick is increasing the wing's camber and effective area. If they only made drag, they would slow the airplane while making it harder to stay airborne; landing flaps do both jobs at once.

Point the nose upward — Not quite. Raising the nose can increase the wing's angle to the airflow, but that is not the special job of the flap panels. Flaps increase the wing's camber and effective area so it can make more lift at lower speed. This is why flap-like devices help gliders, small airplanes, and airliners even when their nose attitudes and engines differ.