Why did American industries grow so dramatically in the late 19th century?Was it the railroads, the new "industrial minds" at work, disinterest in politics or a combination of all three?
American industries grew dramatically partly as a result of the first two factors you mention and partly due to other factors not mentioned here.
During this time period, many new methods of producing goods and of organizing and running businesses were coming into existence. These methods helped industry to grow. One way they did this was by encouraging the growth of railroads. The railroads helped industrialize the country both by creating demand for things like steel and by linking various parts of the country to create a bigger market.
But these were not the only factors. There also political factors. The government was very pro-business and put few limitations on its growth. Just as importantly, there was a huge boom in immigration during this time. These immigrants provided the cheap labor that was needed to drive industrialization.