You can gain levels by battling other aliens. When you win a battle, you gain EXP. When you gain enough EXP, you gain a level. Experience is based on level; if you battle a level 5 you will gain 5 EXP, level 10 you gain 10 EXP, and so on.
Another way to gain EXP is at the Training Center, and it only takes a single click!
You can also lose EXP, this happens when you initiate a battle and lose. The loss is also based on level. If you initiate a battle with a level 5 you lose 5 EXP, a level 10 you lost 10 EXP, and so on.