First, let's begin with an overview.  The Spanish-American War took place in 1898 between the Spanish and Americans over territory and imperialism.  The US intervened in the Cuban War for Independence.  Soon afterwards, the USS Maine was sunk near Cuba, and although we are not completely positive what happened, most assume that Spain retaliated for their loss of Cuba.  During the war, both Cuba and Philippines fought against Spanish imperial war, and ultimately the US won.  The Treaty of Paris (1898) gave the US temporary control of Cuba as well as indefinite colonial control of Guam, Puerto Rico, and the Philippines.  Guam and Puerto Rico remain US territories today.  

The US was able to win the Spanish-American War primarily because of superior naval power.  While the most famous charge was Teddy Roosevelt's Rough Riders charging up San Juan Hill, the naval forces in the Philippines were also very important for American success.  The Battle at Manila Bay in the Philippines was very much one-sided; the Americans had far superior weapons and a much larger fleet.  

In additional to the naval power, the Americans had local support.  Both the Cubans and Filipinos fought alongside American troops to remove Spanish colonial forces and rule.  Having a powerful army, navy, and support from the local population was a major help.

The United States won this war in three ways.

First, the US had a much stronger and more modern navy than Spain did.  This allowed it to destroy Spanish fleets at Manila and in Cuba. 

Second, the US was much more able to supply its soldiers in Cuba.  Cuba was, of course, close to the United States and far from Spain.  US naval power meant that the Spanish on Cuba could not be resupplied.  This helped the US win.

Finally, the US was much more motivated to fight.  American volunteers were very excited about the war while the Spaniards were much less interested.

All of these factors contributed to US victory.

How did the US win the Spanish-American War?

The Spanish-American War was one of the most lopsided victories in American history—and possibly world history. The war begins with Cuba's struggle to rid itself of Spanish rule and the Spanish military's brutal acts against the revolutionary militias. Well-publicized in American newspapers and media, the armed conflict was compared to the American Revolution of more than a century in the past. Increasing public demand for intervention and the USS Battleship Maine's mysterious sinking forced the American government to take action.

Led by Commodore George Dewey, on May 1, 1898, the American naval forces caught the Spanish fleet utterly unprepared for battle and moored in Manila Bay in the Philippines. After destroying and annihilating the Spanish Fleet, American troops turned their attention to finding the remaining Spanish naval forces in Cuba. Adm. Pascual Cervera commanded the remaining Spanish naval forces and had moved them into Santiago Harbor, believing them to be safe from American forces. Cervera miscalculated the speed at which the American troops advanced after landing calvary forces east of Santiago. With Theodore Roosevelt and General William Shafter at command, Cevera was forced to move his fleet away from the harbor and into the open seas.

In the hastily organized retreat on July 3, the Spanish fleet's few remaining ships were hammered by American guns. Again, the once thought of as nearly invincible naval forces of Spain came to a humiliating defeat at the American forces' hands. Most of the ships were beached, burned, or sunk, resulting in Spain's surrender on July 17. In December of 1898, the Treaty of Paris was signed, effectively ending the conflict.