Why is the elephant in "Shooting an Elephant" behaving as it does?

As the narrator explains, the elephant in question is in a period of "must." This is a time when a male elephant's hormones increase, and his testosterone levels shoot up to high levels. This makes the elephant very aggressive and, therefore, a danger to those around him.
Before Orwell's narrator is called to intervene, the elephant had been chained up in anticipation of its period of must. However, it had broken its chain. It had gone on a rampage, destroying a hut, killing a cow, and then, getting hungry, raiding fruit stands and eating the fruit. It had also turned over and damaged a local garbage ("rubbish") truck.
The main point, however, is that by the time the narrator arrives on the scene, the elephant is over his period of must. He is peaceful and no threat. It makes no sense to kill him, but the narrator nevertheless does, despite the cruelty (the animal will die slowly and painfully) and the wastefulness of the act. The narrator goes through with the killing because he feels he must save face.