There are at least two ways in which the 1920s can be seen as a product of Progressive ideals. These have to do with the changing role of women and the impact of Prohibition.
During the Progressive Era, women were very important to the changes being made in society. Many of the reformers who were working to change society were women. This led to some amount of change in society's view of what women's roles should be. By promoting the idea that women could do something other than stay home, the Progressive Era helped to make possible the new freedom that was enjoyed by "flappers" and other women in the 1920s.
A major result of the Progressive Era was Prohibition. Prohibition came out of the reform movements of the Progessive Era, but its major impact was felt in the 1920s. The fact of Prohibition, and the whole "speakeasy," Jazz Age culture that came about as a result of Prohibition, helped to define what the 1920s were in the United States.