To understand this point of view -- which is today recognized as racist -- it helps to consider the historical context. Kipling wrote this poem for the express purpose of persuading the United States to act as an imperial power in the Philippines. Many citizens, like Mark Twain, opposed this. They believed that the Philippines should be left to rule itself.

But Kipling argued that the United States had a moral obligation to take on the duties of a colonial power. As he wrote in a letter to Theodore Roosevelt, the United States had created this obligation when it ended centuries of Spanish control over the islands. The Philippines were in disarray, and Americans bore responsibility for improving things:  

"America has gone and stuck a pick-axe into the foundations of a rotten house, and she is morally bound to build the house over, again, from the foundations, or have it fall about her ears."

What exactly were Americans supposed to build? Kipling envisioned a mix of tasks, some of which we'd recognize today as humanitarian. His poem makes reference to ending hunger and providing people with medical care ("Fill full the mouth of Famine / And bid the sickness cease"). He also alludes to ending warfare among different factions in Filipino society, and creating the infrastructure of civilization, like better ports and roads:

"Take up the White Man's burden, No tawdry rule of kings,
But toil of serf and sweeper, The tale of common things.
The ports ye shall not enter, The roads ye shall not tread,
Go make them with your living, And mark them with your dead."

But again and again, Kipling makes it clear that the native people -- whom Kipling describes in patronizing, racist terms ("half-devil and half-child") -- will resist and resent imperial efforts. He writes sarcastically of how the imperialist will be "rewarded" by "blame of those ye better, The hate of those ye guard."

Kipling's words suggest the analogy of a child being ruled by a parent. The parent toils for the child's own good, but the child doesn't understand that and is ungrateful. But the analogy doesn’t only apply to the relationship between the United States and the Philippines. In the final stanza, Kipling also suggests that the United States isn't quite grown up yet: Americans need to become a colonial power and assume imperial responsibilities. The British might not thank them for doing so, but they will respect them as peers who have taken on the noble burden.

Kipling, Rudyard (1990) The Letters of Rudyard Kipling, Thomas Pinney, Editor. London, Macmillan, Vol II, p. 350.

Kipling's poem "The White Man's Burden" states that it is the responsibility of Britain (and eventually America) to bring European civilization and mores to the developing world.  On the surface, this ethnocentric poem is very paternalistic, as the tone of it states that one must force people overseas to behave like British citizens even if they do not wish it.  This behavior must be forced on them if necessary.  Other than arguing that the poem is racist, one can also make the argument that the British (and American) way of life at that time was not at all great either.  Both countries had poor people who worked sixteen hour days and could not make ends meet.  The gap between the rich and poor was growing, and business at the time did not have regulations and there were few consumer protections.  Women had few rights, and both nations had polluted cities.  While the developing world had its problems as well, Kipling writes as though the Anglo-American world was perfect, when in reality it was not.  

The overall portrayal of indigenous peoples in "The White Man's Burden"is far from flattering, to put it mildly. In fact, it is condescending and deeply offensive. The very title is problematic, for one thing. The vast colonial empire of the West is a framed as a burden, not to the native people being exploited but to the white colonialists themselves. Right from the first stanza, colonized people are infantilized, not fully regarded as adult human beings with the ability to make decisions of their own. As such they are in need of the benevolent guidance of the White Man, who is burdened with the expectations of his race's divine civilizing mission:

Take up the White Man's burden—
Send forth the best ye breed—
Go bind your sons to exile
To serve your captives' need;
To wait in heavy harness,
On fluttered folk and wild—
Your new-caught, sullen peoples,
Half-devil and half-child.

When people have been effectively dehumanized ("Half devil and half child") it is much easier to justify manipulating and controlling them. The poem says indigenous folk are also highly superstitious and primitive. It asserts that if you are not careful all the hard work the colonial powers have done in building their empires will be undone by the natives' laziness and ignorance:

Fill full the mouth of Famine
And bid the sickness cease;
And when your goal is nearest
The end for others sought,
Watch sloth and heathen Folly
Bring all your hopes to nought.

Kipling believes that white colonizers try their best to prevent famine and stop the spread of sickness and disease, but all their selfless work can still come to nothing because these dreadful savages remain mired in lazy superstition.

Kipling never shied away from the unpleasant consequences of colonial rule, but the implication here is that the natives are largely to blame for their own misfortunes. Kipling invites the White Man to take up his burden but not to take upon himself any ultimate responsibility for the natives' welfare.

Take up the White Man's burden—
And reap his old reward:
The blame of those ye better,
The hate of those ye guard—
The cry of hosts ye humour
(Ah, slowly!) toward the light:—
"Why brought he us from bondage,
Our loved Egyptian night?"

As well as being lazy and superstitious, the natives are also portrayed as ungrateful for the benefits of Western civilization. They would much rather remain stuck in their old, exotic ways, blaming all their troubles on their "betters:" their colonial masters who have sacrificed so much to free them from their erstwhile slavery.