Burning adds X damage more to each Fireball attack where X is the number of stacks.
e.g Your fireballs will do 1,2,3,4,5 more...
Second, if you attack someone else other than your current Burning target, the old target will have their stacks "detonate" for X damage.
This also happens if you attack your current target with a physical attack. Whhhbegrl use is that it counts your normal attacks as a "fireball" and it'll raise the stacks instead of depleting it.
Third, burning reduces regeneration by 1 (not sure on the number)
Burning actually seems kind of overloaded to me and I don't think it's very easy to figure it out actually in-game too. I mean, power level wise, it's okay, but rules and design wise... eww.