Why did some believe that a form of socialism was a solution to the economic divide that existed in the United States during the glided age?

Because of two main reasons:

Socialism was on the rise during the late 19th century, which is the same period that corresponds with the Gilded Age. This is because in essence, the most important socialist works, like Marx's the Capital were released shortly before.Socialism in essence is an ideology that promotes equality. The goal of socialism is to end the exploitation of capitalists and give the means of production to workers. Many people in the Gilded Age, believed that these measures would help reduce economic inequality.