As we mention in "Form and Meter" (go check that section out if you haven't already), this poem is as tight as a tick's tutu on a Tuesday. We may have just made that saying up, but the point is that—for all the racist content—we still have a pretty impressively-constructed poem here.
Not only does it have a flawless metrical pattern and impeccable rhyme scheme, but the sounds in this poem also add to its orderly, organized tone. Let's take a closer look, shall we?
Take up the White Man's burden--
Send forth the best ye breed-- (1-2)
Read those lines aloud and pay attention to what your lips do. You should notice that in these first two lines of the poem, we have three beginning B sounds. That repetition of the beginning sound of a word, friends, is what's known in the poetry biz as alliteration—and this poem is positively lousy with examples of that technique. Whether we're reading about a "heavy harness" (5), "fluttered folk" (6), "serf and sweeper" (27), or any other number of examples (see how many you can find), this poem is constantly echoing beginning sounds of words.
But it's also echoing vowel sounds, too. (That's called assonance .) For example, we have the long I sound repeated in "bind" and "exile" (3) and the short I sound in "bid" and "sickness" (20). We even get consonant sounds repeated in the middle of words (consonance), as in the TH sound of "sloth and heathen Folly" (23).
So, what's up with all this sonic repetition? It goes right back to the reason for the poem's tight form and meter. Our speaker is advocating for a kind of order to be imposed on the word, so it makes sense that he does so in a highly-ordered poem. It's as if the lines themselves—and the sounds they contain—are marching along with military precision. That's no small feat, really. It's just too bad this poem pulls all this off in the service of a racist argument.