In 19th century thought, the "White Man's Burden" was a version of noblesse oblige exercised by the 'superior' European races towards 'lesser' races found in Africa and Asia. Whites had a duty to introduce these barbarians to the joys of modern civilization, usually by taking control of their societies and imposing a new order on them by force.
What is truly astounding about the concept of "White Man's Burden" is that many Victorian whites actually believed it. This was not simply a rationalization for them; they thought they were bringing enormous good to barbaric cultures and 'uplifting' races that, in their minds, really were inferior.