You earn it in two ways: you can charge, which initiates a counterattack stance and restores a bit of HP (costs 1 MP), or you can use Channel, which costs 15 TP and gives you one BP and an agility buff.
Elemental fist moves and a couple of high TP physicals will use BP. Noah, the mage, doesn't join until later, so these moves will be key in a few of the battles. Plus, later on there will be enemies with massively high magic defense/reflect but with elemental weaknesses.
Each point of BP (max. 5) adds a multiple to the damage.
So if you have 5 BP, you'll be doing 5x damage than if you used the spell with 0 BP.