A professor emeritus at Bowling Green State University in Ohio, Charles H. McCaghy, has collected photos of these "scandalous" women for a long time. All the photos in this post comes from his collection.
As you can see curvaceous women were preferred before skinny ones. And even if these women hardly showed any skin at all, in their covering tights and long sleeves, it was just the fact that they actually showed off they feminine shapes that was so outrageous.
During the Victorian era women were expected to be modest, conservative, and take care of their homes. The women of burlesque broke completely with these norms, some even donning male outfits on stage. But by exaggerated humor these performers addressed political issues of the day, defying the place society had dictated for women. From the stage they could speak freely and challenge the men in the audience.
When the moving pictures arrived, together with vaudevillian theatre and Broadway revues in the 1920s, burlesque became less popular. With the new competition from all these new forms of entertainment the burlesque dancers had to start taking off more clothes to keep as much of their audience as they could. By the 1930s they were down to pasties and a G-string - which is where we still are in the burlesque of today!