There are two main points in Chapter 8 of A People’s History of the United States. You could combine the two and say that the one overarching main point is that the Mexican-American War was a bad war.
The first point that Zinn is making is reflected in the title of the chapter, “We Take Nothing by Conquest, Thank God.” This is a sarcastic title. Zinn is alluding to the fact that, in his view, the war with Mexico was all about conquest. He says that American elites wanted to take Mexican land and were willing to provoke a war to get it. This was a war motivated by Manifest Destiny, an idea which was based on the idea of American superiority.
The second, and perhaps more important, point that Zinn makes in this chapter is that this was not a war that the American people really wanted. Zinn sees history as being made by elites who pursue their own agendas even as the people try to resist them. This war is no exception. As Zinn says,
It was a war of the American elite against the Mexican elite, each side exhorting, using, killing its own population as well as the other.
Zinn argues that the elites pushed America into the war and that the elites benefited from the war. Meanwhile, the common people did not really want the war. They protested the war and they (those who were in the military) deserted in large numbers during the war. Even so, they were the ones who were made to bear the burden of the sacrifices that the war demanded. In both of these ways, Zinn argues, the Mexican-American War was an ignoble moment in US history.