Whenever a Pokemon does damage to an opponent during battle, the Pokemon will receive Attack Points, equivalent to the amount of damage it did.
e.x. When my Scizor uses Bug Bite and does 64 damage to its enemy, it will gain 64 Attack Points.

Attack Points can be used in the Change Moves page. In order to change a move of a Pokemon, that Pokemon will require Attack Points to purchase the move.
e.x. In order to teach my Charizard Blast Burn, I first need to let Charizard battle and do enough damage to earn the amount of Attack Points it requires to purchase Blast Burn.