O'Connor uses the integrated bus as a sort of petri dish: She mixes up whites and blacks, the poor and poorer, the educated and uneducated, and keenly observes how they interact with another. In that way, the bus becomes a symbol of 1960s South, giving us a microscopic view of a particular slice of society.
The bus is also a place where characters can't hide from each other, especially now that blacks and whites can sit wherever they want. It literally shakes Julian out of his meditations, because every time the bus stops and lets on new people, he is reminded of his present situation. Take the well-dressed black man: is Julian able to have a conversation with him? Nope. What about Carver, or Carver's mother? Not even close.
The bus is a constant reminder that Julian is not so liberal, even if he can't quite see it.