The main reason for the change that you mention here is the New Deal. Specifically, the increase in union membership is due largely to President Roosevelt's policies towards unions. The most important of these policies was the passage of the National Labor Relations Act.
Workers in the United States unionized because they wanted to have better working conditions and higher wages. This is not surprising. The real issue is why they unionized so much during the time that you mention. This is due to the fact that the passage of the NLRA protected their right to unionize in ways that had never been done before. This allowed union membership levels to rise dramatically. As the link below tells us, this law (also known as the Wagner Act)
protected the right of workers to join unions. Workers could no longer be fired for joining a union. It prohibited companies from certain unfair business practices, such as interfering in union activities, refusing to bargain with a union, or gaining control over a labor organization.
Because of these protections, workers unionized in large numbers during the New Deal.