In the essay "Shooting an Elephant," one of Orwell's purposes in telling his story is to show how it gave him a glimpse of "the real nature of imperialism." What does he mean? How does his essay illustrate this purpose?

During his time as a policeman in Burma, which was then a British colony, the story's narrator learns to hate imperialism. As he puts it:

 In a job like that you see the dirty work of Empire at close quarters.

When has to shoot an elephant, the episode exemplifies the irrationality, cruelty, and dehumanizing effects of the system. The event shows him, he says, "the real nature of imperialism," and "the real motives for which despotic governments act."  
When an elephant goes on a rampage and kills a man, the Burmese villagers turn to the narrator to shoot the elephant, as only the British are allowed weapons. By the time the narrator arrives, however, the elephant is calm. There is no need to kill it. In fact, killing it would be a mistake, a waste. However, because of the irrationality of imperialism, the narrator feels he has no choice. He realizes that imperialism robs the rulers as well as the ruled of their freedom.

I was only an absurd puppet pushed to and fro by the will of those yellow faces behind. I perceived in this moment that when the white man turns tyrant it is his own freedom that he destroys. He becomes a sort of hollow, posing dummy.

Shooting the elephant is cruel because elephants are huge, with thick hides, and it takes a long time for them to die. It takes this elephant half an hour to die.
From the start, the narrator shows how imperialism has dehumanized him:

I thought that the greatest joy in the world would be to drive a bayonet into a Buddhist priest's guts.

The narrator realizes as he shoots the elephant that imperialism has stripped his humanity away, making him little more than a "dummy." Saving face and not being laughed at—playing the role of the strong, resolute sahib—has become more important than reason or compassion. Saving face has become the rationale for acting in a despotic regime. The narrator recognizes that imperialism does not work well for anyone. 

When Orwell talks about getting a "glimpse" into the true nature of imperialism, what he means can be summed up in the following line:

A sahib has got to act like a sahib; he has got to appear resolute, to know his own mind and do definite things.

In other words, imperialism forces people to take on very specific roles. Agents of the empire, like Orwell, for example, must act in an authoritarian way no matter what the situation because this is what the native people expect of them. More importantly, this is what imperialism itself expects of them. They must do whatever it takes to maintain their authority and the integrity of the empire.
To illustrate this, Orwell gives a very open and honest account of the dilemma over whether or not to shoot the elephant. Orwell makes it very clear that he has no desire to shoot it. He knows, for instance, that the animal will eventually calm down. However, as Orwell shows through this description of the crowd, they expect him to shoot it:

The people expected it of me and I had got to do it; I could feel their two thousand wills pressing me forward, irresistibly.

This description of the crowd illustrates the essay's purpose because it shows that Orwell must shoot the elephant because the people expect him to do it. If he backs down, he will not only be humiliated, the entire authority of the British empire will be brought into question.

Orwell writes this essay about his service as a policeman for the British Empire in Lower Burma. In this position, he witnesses how the local people resent him as an emissary of the empire, and this helps turn him against imperialism, the system that enables Great Britain to control distant territories and administer them. He sees that the people do not accept the British system, and he also sees that the system pushes people who work for it to act in accordance with its ridiculous principles. For example, Orwell is called to take care of an elephant who is destroying the local bazaar. Though he knows the elephant is harmless, he feels that the crowd of people following him push him to destroy the elephant. He winds up killing the beast to avoid looking foolish in front of the local people. This incident makes him realize that the system of imperialism forces people to take foolish and thoughtless actions because the system implies that Europeans must be in control in the areas in which the empire reigns.