The answer to this can be found in Chapter 11 of the book. In this chapter, Zinn is arguing that the late 1800s were a time when the "robber barons" and other capitalists dominated the United States and ran it for their own benefit. In keeping with this, Zinn says that
The Horatio Alger stories of "rags to riches" were true for a few men, but mostly a myth, and a useful myth for control.
What Zinn is saying is that it was very hard for a poor person to get ahead in the America of this time. However, it was important for the rich to maintain the illusion that poor people could go from "rags to riches." This helped to control the poor people because it prevented them from truly rebelling. They could feel that America had a just system full of opportunity in which they (or their children) could get rich if they worked hard and were a little lucky. This kept the poor relatively content and unwilling to rebel.