How did economic changes transform the American class system and the relationship between classes?

It depends on the time frame of the changes you are referring to.  For more than a century after the beginning of the United States, there was virtually no middle class.  A wealthy elite ran the major levers of the economy and most of government with it, while the vast majority of Americans - in excess of 90% - lived in relative poverty.  When there is that disparity of wealth, there is class conflict, as evidenced by the organized labor and populist movements of the late 1800s.

Changes beginning in the 1920s to the way we paid our workers, the way work was done and the size of our industries created a mostly permanent middle class that only grew over time.  This reduced class conflict, although the relationship between the middle and upper classes is still a bit contentious these days.