Here are several different reasons that supported American imperialism during the late 1800's.
1) Social Darwinism- This concept developed during the late 1800s thanks to the creation of Charles Darwin's book On the Origin of Species. Even though this book studied birds, many citizens applied the concepts in it to humanity. Ultimately, social Darwinism came to represent the idea of "survival of the fittest." This meant that only the strongest and best nations will survive. America used this as a way to justify taking over other lands, as the American government felt that our way of life was superior to other countries/territories.
2) White Man's Burden- This focused around the concept that it was America's job to "civilize" individuals from territories/countries that did not have sophisticated political or social structures like the United States.
3) Economic opportunities- If America was able to gain control of more land, they would be able to obtain more natural resources. Along with this, it would provide American businesses the opportunity to sell their goods to new customers all over the Western hemisphere.