The License Board builds off the Sphere Grid featured in Final Fantasy X. However, in XII, your characters don't only improve through the License Board--they also gain levels from experience points. Whenever you kill a mob, it gives you experience points and license points. You then use the license points to unlock abilities on the License Board. You can only unlock tiles adjacent to the tiles you've already unlocked. Most tiles enable you to use new equipment (like training to use a sword or armor) or spell (there are 5 schools of magic in FFXII). There are also other buffs like +500 HP or reduce time to cast by 10% on the License Board.
5