Take up the White Man's burden-- In patience to abide, To veil the threat of terror And check the show of pride;
Our second stanza starts off much like our first. In fact, line 9 is identical to line 1, making that a refrain. It's clear that our speaker's got a main point to get across here: "Take up the White Man's burden."
So, we know by now what our speaker thinks of white men (they're the best) and the native peoples (they're wild animals), but it's interesting to note here that the speaker sees the White Man's racial and gender superiority as a special responsibility. He's not just cheering on white men for the heck of it. He's saying that, because they are the greatest human beings on the planet, they have a responsibility to help out the rest of the world's inferior, uncivilized inhabitants. And they should accept and tolerate that responsibility with patience ("In patience to abide").
It's the White Man's job to stamp out fear (in another metaphor, "veil the threat of terror") and not to be too proud about his ability to do so ("check the show of pride").
When you put it this way, it almost seems like our speaker is being generous. He's encouraging white men to make a humble sacrifice. Of course, it's hard to look past all the racial insensitivity from earlier, but we wonder if he might be turning a corner here.
By open speech and simple, An hundred times made plain To seek another's profit, And work another's gain.
Yeah, no—that racial insensitivity comes roaring back here as our speaker tells white men to use "open speech and simple,/ An hundred times made plain." In other words, he will have to speak sloooowly and repeat himself a bunch because the natives won't be able to understand him.
(We can't help but wonder here why he doesn't encourage the white men to take a Rosetta Stone course. After all, if they're the best of the bunch, couldn't they easily learn another language? Does our speaker just expect the rest of the world to know English?)
Our speaker concludes with the point that this is all about helping others, but he just can't seem to make a point without insulting a whole race of people.