Why did workers try to organize labor unions in the United States in the late 1800s?

Actually, the first labor unions were founded in the 1830s by women working the mills of Lowell MA and Manchester NH. They were not looking for better pay, they had not thought of that, but of better treatment. After several short strikes, they won.

The Civil War put all social and labor issue on hold for its duration plus a number of years afterwards, the reconstruction era. Starting in the 1870s and continuing into the 1920, industry in the U.S. expanded very rapidly. Naturally, so did the need for labor.

But labor unions arose out of an unlikely labor force. The first organized labor group were the farmers of the mid-west who formed The Grangers. Their prime purpose is what we call PACs today. They would decide as a group which politicians to vote for, those who best supported their interests. The Illinois Granger group successfully won a court case against the railroads who were fixing prices on the commodities the farmers were shipping.

In 1869 Uriah Stephens started a labor union known as the Knights of Labor. They recognized that the growing number of laborers in American mills, railroads, and other large operations needed a voice in what was happening to them. The Knights took the approach of voicing labor's concerns to all industry at once and try to come to a resolution. They were seldom successful though they gain a lot of headlines, mostly negative. The KoL did not have a good answer for strikebreakers (scabs), for the police, for the national guard, or for the private security firms.

Those private security firms were mostly a front for a bunch of thugs who would cause trouble among striking workers by inciting riots, the Haymarket Riot being one of the more infamous. The name most frequently behind those thugs, that also enjoyed much respect of the non-labor citizenry, were from the Pinkerton Agency. Pinkerton had gained the public's trust by reducing train robberies.

But the Knights were not willing to look at laborers individually by job title. Into that arena jumped both the Industrial Workers of the World and the American Federation of Labor. The difference between the two was the IWW took on anyone who would join their ranks while the AFL would only consider white men of skilled labor groups.

The 1912 Textile Strike of Lawrence Massachusetts brought both parties into the forefront each of which contributed to a successful end to the longest and largest strike this country had ever or has ever known within the bounds of a single city. They had a plan to run successful strikes for higher wages and better working conditions. They also found a way to make mill owners accountable, something that had previously been lacking.