During this stage, you determine the damage inflicted by your character's attack, and the Storyteller describes what occurs in the turn. Resolution is a mixture of game and story; it's more interesting for players to hear "Your claws rip through his bowels; he screams in pain, dropping his gun as he clutches his bloody abdomen" than simply "Uh, he loses four health levels." Attacks and damage are merely ways of describing what happens in the story, and it's important to maintain the narrative of combat even as you make the die roll.
Normally, additional successes gained on a Trait roll simply mean that you do exceptionally well. In combat, each success above the first you get on an attack roll equals an additional die you add automatically to your damage dice pool! This creates fatal and cinematic combat.
We refer this extra dice pool as overage. Say George has 5 successes, and Jackie gets 2 to dodge. That is 5 - 2 = 3 that is 2 above the first die. So, you get two dice added to your damage roll.
See Dodging , Soaking , Damage Type , Damage Modifiers and Health and Healing Times for more information.