There are at least two possible ways to answer this question.
First, the intended audience of the poem is the people of the United States. Rudyard Kipling wrote this poem when the US was taking control of the Philippines. Britain (Kipling's home) had long had an empire and Kipling was trying to convey to Americans (who were new to imperialism) what obstacles and difficulties they would face in governing their new possessions.
Second, the poem is aimed at people who are interested in the debate over imperialism. There were those who felt by this time that imperialism was unjust. Kipling's poem is meant to contribute to this debate. Kipling is arguing that imperialism is not a selfish action on the part of the imperial power. Instead, he is saying that it entails selfless toil on the part of the imperial power; toil which is not appreciated by those who benefit from it. Thus, Kipling is trying to argue in favor of imperialism by painting it as a noble and difficult effort on the part of the imperial powers.
In this way, the poem really has two intended audiences.
Kipling wrote the "The White Man's Burden" for the Diamond Jubilee of Queen Victoria (who reigned from 1837-1901), and the poem was published in the New York Sun on February 10, 1899. Later, Kipling rewrote parts of the poem to address the American annexation of the Philippines following the Spanish-American War in 1898. The audience for the poem was the imperialist powers, including the emerging imperialist power--the United States. Kipling was a friend of Teddy Roosevelt, who believed in American imperialism, and the poem was written in part written to convince Americans to take over the Philippines.
Even before the poem appeared in the Sun, parts of it were read aloud in the U.S. Senate to in fact justify not colonizing the Philippines; however, the poem was later read as a justification for colonization. Lines such as "Your new-caught, sullen peoples, Half-devil and half-child" justified the narrative of America's belief in Manifest Destiny and its desire to become an imperial power. While many Americans at the time, including Mark Twain, were opposed to colonization on the basis that it dehumanized and debased conquered people, Kiping's poem justified colonization based on the idea that Americans had a duty to colonize inferior people to bring about their moral uplift.