The cartoon features Uncle Sam and John Bull (the UK equivalent of Uncle Sam) carrying baskets of Africans, Asians, and Middle Easterners over a mountain range. The rocks beneath their feet represent things like ignorance and oppression, while the shining city on the hill represents civilized humanity.
The cartoon is basically saying that non-white, non-Westerners are more animalistic and less rational than white Westerners are. It promotes a colonialist philosophy that dehumanizes the colonized and contracts the idea that the colonizers overtaking these cultures and governments is for these peoples' own good.
Kipling's poem of the same name was written to encourage American colonization of the Philippines, so it is definitely a pro-colonial work. He presents the "civilizing" of the "half-devil" and "half-child" non-white peoples as a hard but noble goal. The "white man's burden" is essentially this colonialist goal and the "new-caught, sullen peoples" are those overtaken by these colonizing nations.
The full title of Kipling's poem is "The White Man's Burden: The United States and the Philippine Islands." Kipling wrote the poem in 1899, the year in which the Philippine-American war began. In that same year, the American Senate passed a bill to place the Philippines under American control.
The poem was written to encourage American intervention in the Philippines and to encourage American imperialism in a broader sense. According to the poem, it is a moral responsibility of the supposedly more advanced white people of America to help the supposedly less advanced, less civilized people of the Philippines.
The political cartoon published to accompany the poem (in Judge Magazine, April 1899) depicts two white men carrying baskets on their backs which are full of African and Asian people. The Africans and Asians are drawn to appear foolish and infantile. The white man leading the way is John Bull, a personification of the U.K., and the white man following behind is Uncle Sam, the personification of America. The two white men are climbing over rocks which are labelled with words like "Oppression," "Superstition," and "Ignorance," and they are heading towards a golden statue named "Civilization."
The message that the cartoon is meant to convey is that it is the responsibility of the Americans to follow the British example and lead the people of the Philippine Islands to civility, just like how the British (so thought Kipling) had led Africans to civility by colonizing parts of Africa. Therefore, the "white man's burden," as Kipling saw it, is the responsibility that the white man has to civilize Africans, Asians, and all "uncivilized" peoples. Anyone of Asian or African descent would be rightfully offended and angry about such a racist message, as it implies that Africans and Asians are less civilized, less intelligent, and more animalistic than their white counterparts.