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 don't want to rely on luck, then the Burst Frogs - when fused will instantly grant one brave burst level up when fused into another unit. There are other Burst Frogs such as the Burst Emperor and Burst Queen to increase a unit's Brave Burst Level.
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. Same goes for when a unit evolves to 7 stars it will acquire Ultimate Brave Burst after it reaches Super Brave Burst level 10 but is only level 1 an not cannot be upgraded further.