Zinn's basic argument about World War I is that it was a war that the United States participated in because of the demands of the capitalist classes.
Zinn points out that the official rationales for war were empty. He says it was not a war about democracy but was, rather, a war about preserving foreign markets for the economic elite of the country. He argues that the war also came about (for the US) because of how closely we came to be tied to the Allies as they bought huge amounts of war materiel from us early in the war. In addition, the propaganda put out in support of the war was, in Zinn's mind, a good way for the government to "create an artificial community of interest between rich and poor." In other words, Zinn is saying the government used pro-war patriotism to shore up support for capitalism by making it seem as if the poor and the elites had common interests.
Therefore, Zinn argues that WWI was a war that was fought for the benefit of the elites in the US.