When you fuse units with the same type of Brave Burst as your Base unit, there is a chance to earn a Burst Level Up. You can check your Brave Burst type on the unit's status screen (View Unit)
The three kinds of Brave Bursts are:
Damage (Anything that deals damage whether AoE or single target)
Healing (Anything that recovers HP)
Support (Boosts all Allies' Def/Crit/Atk/etc)
If you dont want to rely on luck, then the Burst Frogs - when fused will instantly grant one brave burst level up when fused into another unit. They are only obtainable in Frontier Hunter EX as of now.
Also, as of an update in April 2014, when a unit evolves to a 6 star unit, half of its Brave Burst level (rounded down) will carry over to the new unit. For example, a unit with BB Lv. 10 that evolves will start with BB Lv. 5. Another example is a unit with BB Lv. 5 that evolves will start with BB Lv. 2.
When a unit reaches 6 stars in evolution, it will acquire Super Brave Burst after it reaches Brave Burst level 10. Super Brave Burst starts over at level 1 again but is leveled the same way as normal Brave Burst.