How does the elephant die?

In this story, the elephant dies as a result of being shot by the narrator, with a .44 Winchester rifle.
You'll notice from the text that the elephant does not die after the first shot. In fact, Orwell shoots the elephant twice, and neither of these cartridges does enough damage to kill it. While the elephant is clearly in "agony," it takes a third shot to bring the elephant crashing down to the ground. The force of this fall is so great that it shakes the ground beneath the narrator's feet.
The elephant, however, is still not dead. Once it is down on the ground, Orwell fires his remaining two cartridges, finally ending the animal's life.
You'll also notice that Orwell takes no pleasure in shooting the elephant. He kills it only to avoid being publicly humiliated by the native Burmese.