Soon after America found them in the Spanish-American War which America found to be an easy war in fact it only took about four months to beat the Spanish, the result of this war resulted in The Treaty of Paris in 1898. Spain lost more than just a war they also lost Cuba, Guam, and Puerto Rico to America. America was also able to acquire the Philippines during this time, but only Puerto Rico is still part of the American territory today mainly because they have never chosen to declare their own independence.
There are many reason a nation fights for imperialism, however for America there where many reasons. Although the main reason was because America was always connected to the rest of the world and since America was an industrial power they found that they needed to expand the trade market where they could also begin to receive raw materials and overseas areas provided cheap labor which allowed for American goods to be made with a decreased cost. Imperialism just didn't benefit America it also helped the country America was annexing by allowing them to get a pass on the tariffs that America had. Not to mention that this also provided a perfect opportunity for America to instate their idea of democracy to these new territories. America had the belief that granting freedom and a form of independence it would put to rest the domestic unrest that was growing from the lack of growth from their country before annexing into America.