Though labor unions certainly existed long before, the 1930s, under the cloud of the Great Depression, were truly a hotbed for the rise of labor unions and a multitude of other rights groups and organizations fighting for political, economic, and social change. This rise can be attributed to such factors as massive unemployment, unsafe working conditions, low wages, long hours, unsafe and unsanitary living conditions, the loss of life savings, and discrimination of all types.

Now, though it may appear on the surface that all these unions and other groups were fully united in their vision of what America could, or should, be, the reality is that there were deep divisions spelled out along ideological lines. For example, there were labor unions and other organizations that adhered strictly to Marxian economic and political principles and envisioned a radical “do over” of society and economy via revolution. These organizations were emboldened by the Depression, brought on by what they saw as the inherent failures of capitalism, and many of them were further emboldened by the victory of the 1917 Bolshevik Revolution in Russia. On the other hand, there were unions and other groups that took more of a reformist position. Rather than achieving their goals and vision through revolution, which can certainly be violent in nature, these organizations thought it best to work within the confines of the current, or established, system—government, society, industry, and economy—to bring about reforms over time that would improve quality of life.

Zinn’s argument is that the New Deal fell far short in bringing true social, economic, and political change to the United States. Instead, he contends, the New Deal was a program meant to merely placate those individuals and organizations throughout the country who were, in their views, fighting for true change and justice, even if it meant violent revolution. In short, the New Deal, according to Zinn’s argument, did nothing more than keep capitalism in place and the country’s elites in power.

As you say, most histories portray the New Deal as a very important change in government's role in society.  The New Deal is seen as the beginning of the modern state that we have now where government is expected to regulate the economy and to ensure that the economy runs smoothly.  This is seen by most historians (though conservatives tend not to agree) as a good thing.

Zinn, however, has a much more radical view of the New Deal.  To Zinn, the New Deal was simply a way of trying to bolster capitalism and to prevent people from really changing society on their own.  He argues that one of the goals of the New Deal was

...to head off the alarming growth of spontaneous rebellion in the early years of the Roosevelt administration- organization of tenants and the unemployed, movements of self-help, general strikes in several cities.

Zinn portrays the New Deal as an attempt by the government (run of course by members of the elite classes) to maintain control of society so as to prevent a true revolution that would have ended in popular control of the society.