What does leveling an amiibo in Smash 4 do?

As Amiibo level up, their stats will increase. This will lead to Amiibo being statistically more powerful than other players when custom characters are turned off. You can customize how their stats increase by feeding them equipment, but they will increase even if you don't. The total of Amiibo stats (attack + defense + speed) cannot exceed 120.

Amiibo's stats will stop increasing at level 50, although they will continue "learning" and to improve their AI. Amiibo's AI can become considerably better than even a level 9 CPU, and their behaviors will be representative of what they found to be successful, for them as well as their opponents, so far.