Take up the White Man's burden-- The savage wars of peace-- Fill full the mouth of Famine And bid the sickness cease;
Our speaker here gets a bit more specific in how he expects white men to help out. They are to bring peace to the people they conquer—even if it takes a "savage war" to do so. (That, folks, is what you call a paradox.)
As well, they should be helping to eradicate famine. (That's what the metaphor of "Fill full the mouth of Famine" is about.) And they should be stopping ("ceas[ing]") sickness. (Actually, the speaker uses figurative language here to encourage white men to "bid," or ask, sickness to stop. And pretending that sickness has ears and could respond to that request is an example of personification.)
So, after they take over these people and kill a bunch of them in a savage war, white men should give them some food and medicine. Um, thanks?
And when your goal is nearest The end for others sought, Watch sloth and heathen Folly Bring all your hopes to nought.
Of course, this won't be an easy task, our speaker warns (for, like, the ten thousandth time). Just when these humble and hard-working white men are closest to achieving their goal, everything is going to be ruined by the natives.
That's because the local peoples are lazy ("sloth"), uncivilized ("heathen"), and foolish ("Folly"). Wow—stereotype much there, speaker?
Once again, personification helps get the point across here, as Folly (note the capitalization) is given human qualities that might, on its own, undo all the good work that the white men have been engaging in.