The basic answer here is that the nonwhite people get all the benefits of civilization when the white people "take up the white man's burden."
Here are some specific things that the nonwhites get. First, they get healthier -- they learn how to prevent disease -- because the whites can "bid the sickness cease." Second, they can avoid starving because the white people will prevent famines. Finally, the white people will build ports and roads.
Really, though, these are the only specific things mentioned. Most of the poem is focused on telling the white people (America) about how hard it is to try to help the nonwhites -- what a thankless task it is.