There were two main consequences (for the US) of the US victory in the Spanish-American War. One consequence had to do with territory and the other had to do with the status of the United States in the world.
In terms of territory, the US victory in this war led to America taking control over a variety of places. The US took outright control of Puerto Rico. It allowed Cuba to be independent, but used the Platt Amendment to keep effective control over the country. The US also gained possession of Guam and the Philippines.
Perhaps more important than the actual territory was the impact on the standing of the US among the countries of the world. By taking all of this territory, the US became, for the first time, an imperial power. Up to this point, the US had taken control of a great deal of territory on the North American continent, but it had not really ventured overseas much. Therefore, it was not really seen as a major power. By taking an empire, the US took its place among the major powers of the world.
Thus, the US victory in this war gained the US both territory and a more exalted place on the international scene.