Battle experience improves the performance of surviving aircraft and ship crews. The computer AI tracks the battle experience of every ship and every aircraft crew. The best crews are likely to have been both well trained AND battle hardened.
The ship details panel shows current levels of training and experience. In the following picture, the blue line at the top right of the panel shows training levels, and the red bar shows the supplement due to battle experience:
In this example, it can be seen that the British battleship Invincible has a base training level of approximately 4.5, and has gained over half a point from battle experience, making the total crew efficiency just over 5 out of a possible 10.
In the case of aircraft, each surviving aircraft crew gains experience from combat, and that experience is then tracked at the air group to which the aircraft is attached, helping to raise the experience level of the group. (The experience level may also drop (or rise) when new aircaft crews join the group and have a lower (or higher) training level than the current group average).