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

Because socialism in its essence promotes a more equal split and tradeoff between the amount of money and resources that exist in a society. The ideal of socialism would be that everyone would have around the same amount or take the amount he needs and give back the amount he can give, this would then produce a more egalitarian society. This is also the reason why people believed that the United States, a highly capitalistic country, would benefit from the ideas and ideals of socialism.