Why did American cities grow so dramatically in the late 19th century?
Demographics and economics tend to be closely connected, as can be seen in the history of the Industrial Revolution. In pre-industrial, agrarian economies, economic activity was focused primarily on agriculture, especially given that most people lived in the country. As industrialization took shape, however, these economic realities began to change.
Industrial manufacturing requires a concentration of workers in order to adequately function; after all, you cannot effectively run a factory with only a handful of workers. This means that, by its very nature, industrialization tended to be centered in urban, high-density populations where one could find ready access to labor. At the same time, the presence of industry would have led many workers to migrate out of the countryside into the cities, hoping for employment.
When looking at the history of the United States, you can observe this same patter: economic expectations and demographics are closely intertwined. The United States in the late 1800s had entered into the Industrial Age, and the demographic trends reflected this. With corporations centering their operations in urban centers, urban populations grew dramatically as people moved into the cities looking for work and opportunity. Additionally, you should factor in the role that immigration played in shaping these demographics as well (as well as what factors might have drawn them to the United States). Immigration played a key part in this picture.
To conclude, I would suggest that the rise of the industrial economy was the key factor in shaping American urbanization, and that the growth of these cities was a reflection of those transformations.