Take up the White Man's burden-- Have done with childish days-- The lightly proferred laurel, The easy, ungrudged praise.
Our speaker makes one last attempt in this stanza to encourage white men to engage in this work. This time, he uses shame as a motivator (good choice, we'd say).
He tells white men to grow up, stop acting like children, and stop basking in worthless rewards.
A "lightly proffered laurel" is a symbol that refers to a freely-given honor. (The ancient Greeks used to wear a crown of laurel leaves as a mark of distinction.)
Our speaker's message is that it's time for white men to stop bothering with all that nonsense and do something truly worthwhile.
Comes now, to search your manhood Through all the thankless years Cold, edged with dear-bought wisdom, The judgment of your peers!
Now is the time to act, says our speaker ("Comes now"). The white men he addresses must look inside themselves ("search your manhood") and realize that the thankless work ahead of them will pay off in the end.
Not only will they be the recipients of wisdom, but as a result of going through these difficulties they'll also have the good option of their peers as well. And who wouldn't want that?
In the end, the speaker makes a pretty stirring argument for white men to head overseas and spread their way of life around. It's just too bad that this poem never asks the people who lived there in the first place what they think of the idea.