1) The United States acquired vast new territories as a consequence of victory in the Spanish-American War, most notably Cuba and the Phillipine Islands. These new overseas possessions meant that, like the major European powers, the US was now an openly imperialist nation. While there were fierce debates within the US about whether we should expand our reach beyond the North American continent, the reality of American expansionism had arrived.
2) Before the Spanish-American War, the US had a cultural and historical reluctance for standing armies and navies. American civic culture saw large military forces as a threat to democratic freedoms, and most Americans felt very safe with two large oceans separating them from the other Great Powers. Because of the war, the United States vastly expanded its naval presence in the world. Even though the Army remained very small compared to Europeans nations, the US Navy soon rivalled Great Britain and Germany in size and technological capacity. At the turn of the last century, naval power was a sign of influence and stature in the eyes of the international community.
Additionally, having a large navy increased the incentives to show power on the world stage. With a navy, you needed coaling stations, telegraph lines and remote naval bases. Having a navy, that is to say, made it more and more necessary for a nation to expand into the world, and so that also contributed to the US rise to world power.