The two most important assumptions are the assumption that God favors some countries over others and the assumption that some races and cultures are better than others.
Manifest Destiny assumed the superiority of the Anglo-Saxon "race." It assumed the superiority Protestantism over Catholicism and of American culture over other cultures. It also assumed that this superiority was mandated by God. It assumed that any country or any race that could be so clearly superior must be favored by God. This added a moral dimension to the argument. Manifest Destiny was not simply saying "might makes right." Instead, it was saying that the mighty have the moral right (and even duty) to conquer the others because they have been favored by God.