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

Answer: Make the wing more curved

Make the wing more curved ✓ — Right. Flaps change the wing into a more curved, lift-friendly shape and often increase its effective area, so the same speed can make 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 make more drag — 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 changing the wing's lift-making shape. 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 change the wing itself 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.