Ships and planets get experience when doing things. Experienced units are better than inexperienced ones in certain aspects. In a nutshell:
- Experienced units fight better
- Experienced ships have a smaller risk of running on a mine
(4.0i) Ships can be given hull functions that require a particular experience level to work.
- Ships gain experience by moving, doing alchemy, fighting, and training
- Planets gain experience by fighting
- Just sitting there and doing nothing also gets you a little bit experience
(4.0i) Academy ships can educate new crew faster than normal ships
Full details are on the Experience page.