On page 203 in my edition of the book, Zinn quotes a New York Times editorial that says that denying blacks the right to vote was necessary "under the supreme law of self-preservation." The idea behind this was that giving blacks any sort of power would be dangerous to white people.
This idea was based on the assumption that blacks were dangerous to white people. Right after the quote above, Zinn talks about the ways in which whites saw blacks as animalistic. Blacks were wild things who could be counted on to attack whites whenever they were given the opportunity.
If whites saw blacks in this way, their reasons for denying blacks the right to vote are more understandable. Whites felt that blacks were inherently dangerous. Giving them the right to vote would give them more of a sense of power and make them more likely to harm whites. Thus, denying them the right to vote was an act of self-preservation.