The claim that the United States had this manifest destiny was based on the fact that the US was the richest and most powerful country in the Western Hemisphere by the mid-1800s when the phrase was introduced. By the 1840s, the US had achieved this status. It was also the only truly stable democracy in the hemisphere (since Canada was still a colony of Britain at this point). There was, to be sure, a component of racism in this sentiment as well -- there was the idea that the US was rich and prosperous because of the superiority of its governmental system, its religion and its racial background.
Because of these factors, Americans felt themselves to be superior to the other people and nations in their region. They felt that this superiority entitled them to expand their borders in a manifest destiny.