"The White Man's Burden" is a poem by Rudyard Kipling published in 1899. British imperialism around the world, including India, was already in full swing by then, so we can't say that Kipling caused British imperialism.
But if you read the poem, there's something very jarring about it, at least to a modern cosmopolitan reader: It is unabashedly, even proudly, racist and imperialist. It was written in response to the US invasion of the Philippines, and the story it tells is not of a powerful country conquering and oppressing a weak one, but of a superior, advanced culture trying their very best to civilize and improve an inferior, primitive one, and being constantly thwarted by the primitive savages. It isn't exactly rejoicing in imperialism, but the downsides it talks about are all about how hard it is being the White imperialist, constantly misjudged for his altruistic actions. The desires of the people being conquered are conveniently ignored.
Thus, we can think of "The White Man's Burden" as a kind of apologia for imperialism, and particularly focused at American and British readers. That word is sometimes translated "apology", but "apology" sounds like you're saying sorry for something you agree is bad; an apologia is a defense of why it is not bad, even though so many people seem to think so. We can't really know what Kipling himself believed (perhaps it was intended as satire?), but if the poem is read literally, it is a staunch defense of imperialism as the necessary burden of White men to bring civilization to the primitive masses, whether they want it or not.
This was of course a narrative that the British used heavily, in an attempt to justify their imperialism in India, Africa, and around the world. "It's for their own good" makes the violence and exploitation much easier to swallow than it would have been if they'd admitted it was mainly for the gain of the rich and powerful in Britain. And they had just enough examples of actual improvements in standard of living (accomplished in the most violent way possible, of course) to back up their argument; yes, there was in fact real economic development in India under the British Raj. Defenders of imperialism conveniently ignored of course that there could have been much more development had they engaged in fair and equal trade rather than imperialist exploitation---but there was in fact some development, which gave them some measure of excuse or deniability, and many in Britain clung to these excuses and spread them widely. Kipling's poem is part of that attempt to give an ethical and altruistic veneer to a project that was at its core violent and exploitative.
This soon backfired, however, as "the White Man's Burden" actually became a satirical slogan against imperialism in the US and Britain (and is sometimes still used that way today). It was used as an example of how weak and inhumane the justifications for imperialism truly were.