Most predators have a well delimited territory.

Inside their territory, they will attempt to kill any other predator who could compete with them and who belongs to a weaker species. This is a necessary strategy, because any territory has a limited productivity and it cannot sustain too many predators that want to eat the same kind of prey. Thus predators either specialize into separate niches, e.g. some eat mice, some eat rabbits and some eat deer, or they kill each other if they want the same food, to eliminate the competition.

They will also attempt to repel outside their territory any predator of the same species with them. They will seldom attempt to actually kill a predator of their own species, but that mainly because this would be risky, as in a fight to death they could be killed themselves, so ritualized harmless fights are preferred.

The difference with some primates like chimpanzees and humans, is that competitors of the same species may be treated as other predators treat only predators from different, weaker species.

The reason might be that when you cooperate within a bigger team, you may have the same advantage against competitors that a stronger predator has against a weaker predator, e.g. a wolf against a fox.

Thus a fight to death may be chosen, because the bigger team has good chances to win the fight. So chimpanzees start wars for the same reason why Russia attacks Ukraine or USA attacks Iran, those who have more weapons and more money believe that they can win the war, so they start it.

Most other predators do not start wars against their own kind, because in a balanced fight the winner is unpredictable.