Why are the controls opposite/inverted when flying in a game?
In a game, we would normally use the directions in a normal way to move, i.e. "up" for move forward, "down" for move backwards, "left" for left, and so on.
Although, when we play a flying game, why are the controls opposite, so "up" is down, and "down" is up, "left" is right, and so on.
How does this effect come naturally when we play it?
Many people play First-Person Shooters (or any game in the First Person perspective) with the vertical axis controls reversed. It's all down to personal preference. For me, it feels like pushing the head/gun forward to look down and pulling the head/gun backwards to look up.
Imagine the joystick comes out of the top of the plane. What direction do you move the joystick to make the plane point downward?
The left and right are usually not inverted, I've yet to see an example where someone inverts them.
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
Also, many aircraft do have a control stick, not the wheel looking column. Jet fighters, ultra lights, helicopters for example.
@AustinFrench However, the control stick works in effectively the same way - push forward to dive, pull back to climb.