One reason that so many people participated in politics in the late 19th century is that Democrats and Republicans were almost evenly matched in claiming the national electorate. Elections were decided on very thin margins, so it was essential to vote. About 80% of the electorate voted in elections, which is much higher than the percentage of the electorate that votes in most elections today (see the source from Sage American History at the link below). Shortly after Reconstruction, which lasted from 1865 to 1876, many African American men voted; they were disenfranchised over time, making the voter turnout higher in the years during and right after Reconstruction.
In addition, politics on the local level played a very large role in people's lives. While the federal government did very little, the politicians in a city or town were responsible for controlling local services and even doling out jobs through patronage. The national Congress did very little, but people needed services such as fire departments, police protection, and garbage services to cope with their lives, which were increasingly urbanized. In addition, during this time, cities such as New York City built waterworks to provide fresh water to their populations, which were faced with insecure or contaminated sources of water before that point (see the source by Melosi, below). Therefore, voters were motivated to become involved in the political process to ensure that local governments would provide these vital services to them.
Martin V. Melosi, The Sanitary City: Urban Infrastructure in America from
Colonial Times to the Present. Baltimore: Johns Hopkins University Press,