The description just says that you get extra strength in the face of "inevitable death". I've tried testing it, but it doesn't seem to activate at low health or during second chance. Considering that Killer Instinct already increases damage when at low health and Second Chance already handles your damage during that, I have no idea what Revenge could be for.

EDIT: Maybe it activates from taking a certain amount of damage within a certain amount of time? That's the only unique effect for it I can currently think of.

EDIT 2: Even when losing most of my health in a matter of seconds, I didn't see any indication of the perk activating. I've heard the perk that decreases death penalties is bugged, so maybe this one is too? I wish the game actually explained what it does.

EDIT 3: I finally found out what Revenge does. When you're going to die, it gives you a few seconds to keep fighting before death. It only activates when you're actually going to die, so entering Second Chance doesn't activate it, but failing to get back up from Second Chance does.

Good to know. I unlocked this feature last night after finishing a DLC mission so this will come in handy for those tough boss fights.