In an actual plane, the steering column isn't really pulled up or down. Instead the pilot pushes the steering wheel or joystick away to dive, and pulls it towards him to pull up. Same goes for games which you control with a joystick. You push it away to go forward, and back to go backward.
This forward and backward motion is most probably why inverted controls are intuitive for flying in a game. Leaning forward makes your body face towards the ground , leaning backward lets you face upward. This corresponds to the mental model we have of flying a plane and the associated tilting of the plane itself.
So why have the up and down keys inverted? Arrow Up maps to a forward motion of the joystick/yoke, which maps to a dive in an airplane. Arrow down maps to a backward motion of the joystick/yoke, which maps to pulling up.
Edited the answer to incorporate some of the comments made by GalacticCowboy and Austin French