The answer to this can be found in Chapter 8. There, Zinn argues that the war with Mexico was a war of conquest that was undertaken against the will of many of the American people. The war was fought mainly for the purposes of expansion.
Zinn does not really dispute the most common US motives for engaging in this war. Mainstream historians today tend to argue that the US went to war in order to help make itself more powerful and wealthy. Zinn sees things in much the same way. He says
Accompanying all this aggressiveness was the idea that the United States would be giving the blessings of liberty and democracy to more people. This was intermingled with ideas of racial superiority, longings for the beautiful lands of New Mexico and California, and thoughts of commercial enterprise across the Pacific.
What tends to set Zinn apart from mainstream historians is his account of public opinion with regard to the war. Zinn argues that the war was not supported by most Americans. He cites things like high numbers of desertions from the army. He notes that there was a drop in the number of men enlisting to fight after an initial rush of patriotism. Zinn acknowledges that it is very hard to know what public opinion was in those times, but he argues that many people opposed the war.
Thus, Zinn would say that the US took the Southwest to enrich and empower itself and that it did so over the objections of many of its citizens.