This question makes two assumptions: firstly, that the United States is superior to other nations, and secondly, that this superiority was caused by changes in foreign policy after the Spanish-American War. Both assumptions are subject to debate.
Yes, the United States has been the world's leading economy, but it had already become the world's largest industrial economy (surpassing Great Britain's) decades before the Spanish-American War. It also had a reputation for non-interference in the sovereign affairs of other nations and for avoiding foreign wars. One could argue that that was the real American greatness, before over a century of imperialism, foreign military adventurism, unprecedented expansion of Federal power, high taxes, and record levels of debt resulted from endless foreign wars.
Be that as it may, here are four potential topic points:
The United States, it was argued, must make the world safe for democracy by extending its Manifest Destiny beyond its borders. This argument was trumpeted to the American public by yellow journalists like William Randolph Hearst, because lurid tales of alleged Spanish atrocities in Cuba (ignoring the acts of terrorism by the separatists) sold more newspapers than honest reporting. It also supported the new "large" foreign policy of advocates of imperialism, like William McKinley, Theodore Roosevelt, and Henry Cabot Lodge. Woodrow Wilson famously became a leading advocate of this moral interventionist line of reasoning, showing that it crossed party lines.
The United States must seek new markets and greater market access for our capital investments abroad (i.e., American economic security requires and justifies imperialism). Some nations unfairly closed off their markets to foreign capital investments or foreign imports, and these backward nations must be integrated into the world economy for their own good and for the good of the world (the argument goes).
The then-popular theory of Social Darwinism was developed when Herbert Spencer applied Charles Darwin's evolutionary theory to human affairs and argued that, because nature was a struggle resulting in the survival of the fittest (those that adapted best to the current environment), the United States had a right to assert its struggle for its interests around the globe, like other nations were doing.
"Civilizing" the world and bringing an end to barbarism, slavery, oppression, and tyranny was held to be a moral duty. Was it not manifestly true that the world was full of barbarism, slavery, oppression, and tyranny? Would it not be morally irresponsible for the United States, with its unmatched power to do good, to isolate itself instead of fighting for the oppressed around the world?
These were the main points used to justify American imperialism and intervention in foreign wars (a complete reversal of American foreign policy since 1789).