In the story,Animal Farm,Why do the animals start killing each other?

It's more like Napoleon kills other animals to secure his power.

some of the animals recall the Sixth Commandment, “No animal shall kill any other animal.” Clover again asks Muriel to read to her from the wall, only to find that the Sixth Commandment has been changed to: “No animal shall kill any other animal without cause” (98). This enables Napoleon to systematically cary out executions. He secures his power through fear and intimidation.

Napoleon convenes all of the animals in the yard. With his nine huge dogs ringed about him and growling, he stages an inquisition and a purge: he forces certain animals to confess to their participation in a conspiracy with Snowball and then has the dogs tear out these supposed traitors’ throats. The dogs, apparently without orders, even attack Boxer, who effortlessly knocks them away with his huge hooves. But four pigs and numerous other animals meet their deaths, including the hens who rebelled at the proposal to sell their eggs. The terrible bloodshed leaves the animals deeply shaken and confused. After Napoleon leaves, Boxer says that he would never have believed that such a thing could happen on Animal Farm. He adds that the tragedy must owe to some fault in the animals themselves; thus, he commits to working even harder. Clover looks out over the farm, wondering how such a glorious rebellion as theirs could have come to its current state. Some of the animals begin to sing “Beasts of England,” but Squealer appears and explains that “Beasts of England” may no longer be sung. It applied only to the Rebellion, he says, and now there is no more need for rebellion. Squealer gives the animals a replacement song, written by Minimus, the poet pig. The new song expresses profound patriotism and glorifies Animal Farm, but it does not inspire the animals as “Beasts of England” once did.