Note the phrase "oath or affirmation"; unless I miss my guess, I'd imagine they're referring to the Quaker practice of refusing to take oaths, but instead to "affirm" that they're telling the truth. In any case, it certainly also gives atheists equal status with religious people in this context.

This seems to open some big, wide holes in the theory that the framers of the Constitution intended to confer some kind of special status on fundamentalist Protestantism.