Adrenaline, opiates, and metabolism: So assuming we want this to be biology, not magic, we need to break this down into its components. You have a creature that becomes strong and feels no pain during fights, and is able to heal quickly if it kills/harms enemies. An explanation for regeneration is a bit of a separate question, so I'll focus on the whys more than the hows.

  • The natural fight-or-flight response is geared to allow animals to do things in a time of stress they couldn't possibly do when stress free. People really DO lift cars off children, and leap up trees, because of the ability to increase the percentage of muscle use. The incredible hulk is over-dramatization more than pure fiction. OF COURSE you creature is stronger and faster in a fight, but it's response is highly evolved. I would also suggest these things probably overheat quickly and collapse in exhaustion once the fight is over.
  • The same responses control the production and release of opiates. OF COURSE your creature feels no pain, because fighting is like shooting up with heroin to it (literally, as the drug mimics the brain chemistry).
  • I'll assume your creature heals pretty well regardless of fighting, but it will need a very efficient digestive system, or possibly a bypass that allows it to incorporate proteins somehow directly from its enemies. Maybe it even has a way of interconnecting it's circulatory system and nervous system with other critters. You critter is eating its enemies and fueling a wild metabolic regeneration process. Some kind of immunological shunt lets it incorporate partly finished components to enable rapid healing and growth. Maybe your critter is even stealing the body parts from its enemies and replacing its own lost pieces. Over time, it breaks down the stolen parts from the inside and replaces the foreign cells with stem cells that migrate in.
  • The immunology of this process would be really challenging to justify. Vital organs should still be critter-only so you could "dispose" of the foreign tissue if it was a problem without dying. Usually your immune response will attack foreign tissue, so in this scenario your critter would work best fighting new kinds of enemies it hadn't developed a response to yet. In an alternative scenario, your critter has adapted to incorporating the parts and proteins of known enemies, and the rules are reversed: You heal more from enemies you have fought before because your body already knows what to do with them. Unfortunately, in either scenario, your creatures will be plagued by arthritis, cancers and parasites due to their ability to suppress the immune response to foreign proteins and the long-term responses to the foreign proteins.
  • This functionally would be a lot like Achilles choice - a long and uneventful life, or a short life of glory. The more you incorporate foreign material, the more you need to suppress immune response, and the less you incorporate, the less cancer and disease you suffer. Dying of old age would be a humiliating thing for a warrior, and culture would be passed on by females who would need to stay healthier to raise children.