What motivated the US to try to possess its own empire during the 19th century?
I think that the desire for empire in the late 19th Century American thought was a reflection of the wealth that was being generated. In the end, America was producing a great amount of wealth. It is from this point that thinkers like Howard Zinn reflect that compelled America to want to generate more wealth in foreign markets. Zinn argues that those in the position of economic power felt that the opportunities afforded to them could not be simply contained in America, and necessitated expansion across the world. The development of new and vibrant foreign markets that would generate even more wealth should be the driving force in political policy around the world. When President Roosevelt, "welcomes war," it is with this idea in mind:
The profit system, with its natural tendency for expansion, had already begun to look overseas. The severe depression that began in 1893 strengthened an idea developing within the political and financial elite of the country: that overseas markets for American goods might relieve the problem of underconsumption at home and prevent the economic crises that in the 1890s brought class war.
This is one potential explanation behind the motivation of the United States to possess its own empire during the 19th Century.