What actually happens when an Amiibo levels up and what causes it to level up?
According to Super Smash Bros wikia:
Amiibo can be used as a specialized computer player, with the tag "FP" (Figure Player) above them.
Figure Players start at level 1 like regular CPU players, but as they fight more often, they can continue to level up, to a cap of 50.
As they gain levels, amiibo can potentially "learn" techniques from other characters: if a player frequently uses long-ranged, camping techniques, then the amiibo will also begin to emulate this behavior. This learning continues even after the 50 level cap is reached.
In addition, an amiibo can gain improved fighting statistics, such as improved damage, knockback, and reaction times, the higher its level is. Finally, higher-leveled Amiibo can adapt to individual matches faster than lower-leveled Amiibo.