There were many different reasons that people supported imperialism. I have divided some of them into three categories: economic, ideological, and strategic.
Economic: Many (but not all, Andrew Carnegie was anti-imperialist) business leaders demanded that the United States government acquire secure markets for their manufactured goods. They feared that domestic markets would not be enough to sustain continued profits and national economic growth. So they sought to expand American influence to places like Asia, especially China. The desire to gain inexpensive raw materials for industry, as well as labor, was another factor.
Ideological: Americans of the preceding generations had cheered America's "manifest destiny" to expand and conquer the entire continent. Many Americans by the end of the century thought that they, as a democratic power, had the right, even the duty, to join the other powerful nations of the world as an imperial power. Many also believed that they had an obligation, a "white man's burden," to bring the blessings of civilization to people around the world. Others justified imperialism according to the tenets of Social Darwinism, which claimed that human existence was a struggle, and the strongest people (and nations) would flourish. Finally, Christian missionaries clamored for the chance to spread their faith to people everywhere.
Strategic: Many Americans had become convinced of the importance of naval power to America's emergence as a first-rate nation. A book published in 1890 by Admiral Alfred Thayer Mahan entitled The Influence of Naval Power Upon History was very influential in this line of thinking. Many American strategists demanded that the United States secure naval bases and coaling stations around the world to project American naval power. Also, by this point the nations of Europe had divided up Africa, and were in the process of carving out spheres of influence within China. The United States would lose all chance at becoming a global power if they failed to jump in the imperialist game while they still could.