What is the "exile" that Rudyard Kipling is talking about in the poem "White Man's Burden?"
Kipling says that Americans should "bind their sons to exile/To serve your captives' need." By "exile," he means that America's young men, especially its soldiers, ought to take up the challenge of bringing "civilization" to the Philippines, the colony that the United States had recently acquired from Spain after the Spanish-American War.
Men like Kipling (and notably Theodore Roosevelt) imagined imperialism as a sort of test of masculinity for young men. It was their "burden," being in Kipling's estimation culturally and racially superior, to spread the benefits of civilization to "primitive" people around the world. So by "exile," Kipling meant something like "mission." People around the world, he thought, would benefit from colonization, even if they didn't realize it, and the youth of powerful nations ought to take up the challenge of bringing it to them.
But his crucial point, and the one most relevant to the question, is that the colonizers themselves, through the struggle and strife that accompanied bringing conquered peoples into an empire, would emerge stronger, more moral, and with the respect of their peers if also the hatred of colonial peoples. So Kipling thought that going to faraway places and carrying civilization with them would improve America's "sons."