EXP, or Experience Points, is a stat in the game. It allows characters to level up, making them grow stronger in battle.
Every time you successfully hit an enemy with an HP Attack, you gain a proportionate amount of EXP to the damage dealt. There are certain ways to increase the EXP you gain after each HP Attack. Examples are playing on the "Bonus Day" in the Calendar and equipping the Chocobo accessories obtained after you finish a Play Plan.