What happens if a air plane flies into space

Read up on the X-15 rocket plane program. It was a very fast aircraft that went so high it may as well have gone into space. One thing that happens in space is that the wings and rudder are useless because there is no air to push against. So steering jets were installed to control the tilt of the plane. Then it entered the air again at a high speed and heated up from friction. Finally it landed in the desert like a big glider.

First, let's thnk about what you mean by "fly into space". I thnk you mean at least a sub-orbital hop. This is the only option. And here is why. If it is a regular jet engine or propeller plane then it is unlikely it will hop out of the ozone layer. Even if that happens, the engines will flame out because of lack of oxidizer(oxygen), without an RCS control system it will be impossible to control its attitude until it comes in contact with at least a bit of air. Also, the air pressure difference might do something bad.
If it were a special plane it could even go into orbit. It would have enough thrust to gain enough speed while in the atmosphere, better intakes, a heat reduction system, an air tight system, an RCS system and rocket engines with liquid oxygen oxidizing the fuel to keep it burning. The rocket engine is needed for orbit circularisation.

An aircraft (as the name implies) relies on air:-

- it needs air passing over its wings to generate lift
- it needs air passing over its control surfaces to maneuver
- it's engine requires the oxygen in the air to operate and provide propulsive power

So as it gets higher and higher in the atmosphere and the air becomes increasingly thin all those functions cease to operate. The precise altitude that this happens varies with the design of the aircraft and its engine which is why different aircraft have different maximum altitudes.

The problem with the engine can be overcome by using a rocket engine instead (rockets provide their own oxygen in their fuel) but the basic problems of lift and maneuver remain. It would become an uncontrolled projectile shooting through space until the engine was cut off - at which point it would descend back into the atmosphere and either possibly burning up as it did so or until the atmosphere became dense enough for control to be restored.