Kipling begins the poem by telling the readers that they must "take up the White Man's Burden." This "Burden" is the responsibility that the speaker thinks white people have to civilize non-white people.
The poem was written in 1899 at the beginning of a war between the US and the Philippines. Kipling wrote the poem to encourage Americans to push ahead in their efforts to conquer and colonize the people of the Philippine Islands. In the poem, Kipling describes these people as "sullen peoples, / Half devil and half child." Kipling assumes that it is the responsibility or "Burden" of white Americans to tame and educate these supposedly "half devil and half child" people. This is of course a deeply racist attitude.
When Kipling asks his American readers to "send forth the best ye breed," he is encouraging American families, and, more broadly, the whole country, to send the best men available to help win the war against the Philippine Islands. To the same end, the speaker tells readers to "bind your sons to exile," meaning that families should send young men abroad, to the Philippine Islands, to help win the war for America.
Throughout the poem, the speaker portrays as a solemn and honorable duty the responsibility of white people to civilize non-white people. He says that it is the responsibility of white people to "bid the sickness cease" and to "veil the threat of terror." In other words, white people must cure the "sickness" that is synonymous with non-white skin, and combat the "terror" consequent of uncivilized non-white people. This is a task which, the speaker says, only "the best ye breed" can take up.