"Universalism" is his term for his own position. He does not offer a full account of this position, but he associates it with "the great thinkers of the Enlightenment" (especially Kant, one assumes) and the "fundamental moral equality of human beings" is central to it. He appears to believe that this central principal entails support for open borders (or, as he might prefer to put it, freedom of movement and association), because "the welfare gains that would come from even a mild decrease in coercive limits on travel and free association are awesomely huge." In other words: Latin American immigrants gain more from crossing the border into the U.S. than native Americans lose (through depressed wages, increased welfare spending, culture shock, or what have you). Since all people are morally equal, the moral calculus favors opening the borders. It is wrong for Americans to favor the welfare of their fellow Americans (let alone their own individual welfare!) over that of others.
"Chauvinism" is his term for his opponents. He describes the "chauvinist" position like this: "for the chauvinist, if a rule or policy benefits the group of which the chauvinist happens to be a member, then it is justified." So "chauvinists" oppose open borders, because they care more about relatively small losses to themselves and their countrymen than they do about relatively large gains for non-countrymen - which violates the principle of the "fundamental moral equality of human beings," thereby "getting morality fundamentally wrong."
I hope that was a reasonably fair summary of WW's views.
Before I move on to Part II, I'd like to hear from any colleagues & commenters here who might be interested: what do y'all make of WW's argument?