Enrage is a triggered ability on creatures that fires whenever that creature is dealt damage. The trigger reads "Whenever this creature is dealt damage, [effect]." The effect varies—drawing cards, dealing damage, making tokens, or pumping itself.
The key nuance: Enrage cares about damage being dealt, not about surviving. If your creature dies from combat damage, the Enrage trigger still goes on the stack and resolves—it doesn't matter that the creature is gone. Also, any source of damage works: combat, burn spells, fight effects, even your own pinging. People often forget you can trigger your own Enrage creatures deliberately for value.
In Commander, Enrage shines with effects that deal small amounts of damage to your own creatures—pingers, fight spells, or wither/damage-based board states. It also punishes opponents for blocking or attacking, making your creatures awkward to interact with in combat.