This poem was written in the wake of Spain ceding the Philippines to the United States as a colony. The title refers to the "burden" white people take on when they colonize countries largely populated by darker skinned people.
This poem, clearly racist, has become notorious as a statement of the ruling class point-of-view on colonialism. From the perspective of the victorious colonizers, they are bringing "civilization" to "primitive" people. Kipling is decrying the fact that the colonized don't appreciate the many gifts the colonizers have to offer or the sacrifices they are making to bring civilization to the natives. He instead characterizes the native people of colonized countries as childlike and animalistic, with a sullen, violent, and hateful attitude toward their colonizers. As the word "burden" indicates, Kipling makes the overlords into the victims of colonization—but victims who bear the torture of colonizing other countries with nobility and stoicism.
Kipling slides over the fact that the colonizers did vastly more taking than giving, and that some might characterize the taking as akin to looting. He also ignores the fact that in most colonized countries, the colonizers were not invited and not wanted. The Philippines, for example, wanted freedom and independence, not a new overlord in the form of the United States. Most native people also were not terribly interested in the "civilization" that the West had to offer, being perfectly happy with their own cultures. One might argue, from the perspective of the colonized, that anger is an appropriate response to invaders sweeping in, taking over your country, robbing you of your resources, telling you that you are ignorant savages, and forcing you to adopt a culture you don't want. Kipling apparently didn't look at it from this point of view.