How did the philosophy of "The White Man's Burden" develop into an American attitude of superiority?
I would argue that the idea of the "white man's burden" reflects an American attitude of superiority rather than causing it or developing into it.
The idea of the white man's burden is already one of superiority. When you say "we need to go out and civilize the other people of the world," you are implying that you are superior to them. You have something to teach them and they have nothing to teach you. If you have that attitude, you already have a feeling of superiority.
I would say, then, that the attitude of superiority came first. White Americans already felt superior to non-white people like the Filipinos. The philosophy of the white man's burden developed as a way to use that superior attitude to justify imperialism.