I think that DuBois uses the concept of "the veil" to make the distinction that there is a line or border that demarcates the world of White America from that of Black America. While the theoretical conditions of the Constitution might say differently, DuBois is very much animated by the idea of a world being "veiled" from White Americans, and one that is fully understood by being Black in America. The veil is an interesting image because it implies that there are individuals who are inside of it, able to perceive and see the world with some filter, while there are those outside of it who are unable to perceive little, if anything, about those inside of the veil. In this light, the veil is something that separates both realities, feeding his idea that the fundamental point of division is that of color in America:
'How does it feel to be a problem?' asks DuBois, referring to the conundrum of black identity. DuBois’ first encounter with his status as a “problem” takes place in school when a little girl refuses a card he has offered her as part of a class-wide card exchange. He realizes that the Negro has been taught to view himself through the eyes of others and lacks another source upon which to base identity. This results in a “veil” between the black man’s world, where identity is constructed for him, and the white world, where there are more opportunities and possibilities.
It is the veil and being on the inside of it and looking out that feeds into what DuBois calls "the double consciousness" of Blacks in America, as they are aware of what happens to them inside the veil, but are also trapped in that they know another world, probably one with more advantages, exists outside of it and there is little they can do about it. It is interesting to also note that the use of "veil" also means that the outside world is left to not wonder that much about what is happening inside "the veil," reflecting another unequal reality in the modern setting.