Basically… It was the idea that “The Christian God gives us providence over this whole continent.”
They were of the opinion that they were doing the “Native” GOOD by destroying their culture because their god(s) was “Wrong” or “Evil” in some way and that by bringing their culture and making it the ONLY ACCEPTABLE CULTURE, they were “Saving” the natives.
http://en.wikipedia.org/wiki/Manifest_Destiny
Manifest Destiny is the historical belief that the United States was destined and divinely ordained by The God of Christianity[1][2][3][4][5][6][7]to expand across the North American continent, from the Atlantic seaboard to the Pacific Ocean. Sometimes Manifest Destiny was interpreted so widely as to include the eventual absorption of all North America: Canada, Mexico, Cuba and Central America. It has also been used to advocate and justify other territorial acquisitions, as well as to justify the genocide of the Native American populations who were standing in the way of its believers and supporters. Advocates of Manifest Destiny believed that expansion was not only good, but that it was obvious ("manifest") and certain ("destiny"). Originally a political catch phrase of the 19th century, "Manifest Destiny" eventually became a standard historical term, sometimes used as a synonym for the expansion of the United States across the North American continent which the belief inspired or was used to justify.
The term was first used primarily by Jacksonian Democrats in the 1840s to promote the annexation of much of what is now the Western United States (the Oregon Territory, the Texas Annexation, and the Mexican Cession). It was revived in the 1890s, this time with Republican supporters, as a theoretical justification for U.S. expansion outside of North America. The term fell out of usage by U.S. policy makers early in the 20th century, but some commentators believe that aspects of Manifest Destiny, particularly the belief in an American "mission" to promote and defend democracy throughout the world, continues to have an influence on American political ideology.[8]
Manifest Destiny was always a general notion rather than a specific policy. The term combined a belief in expansionism with other popular ideas of the era, including American exceptionalism, Romantic nationalism, and a belief in the natural superiority of what was then called the "Anglo-Saxon race". While many writers focus primarily upon American expansionism when discussing Manifest Destiny, others see in the term a broader expression of a belief in America's "mission" in the world, which has meant different things to different people over the years. This variety of possible meanings was summed up by Ernest Lee Tuveson, who wrote:
A vast complex of ideas, policies, and actions is comprehended under the phrase 'Manifest Destiny'. They are not, as we should expect, all compatible, nor do they come from any one source.[9]
The concept of Manifest Destiny has acquired a variety of meanings over the years, and its inherent ambiguity has been part of its power. In the generic political sense, however, it was usually used to refer to the idea that the American government was "destined" to establish uninterrupted political authority across the entire North American continent, from one ocean to the other.
Manifest Destiny and the U.S.-Mexican War: Then and Now
http://www.pbs.org/kera/usmexicanwar/educators/md3_war.html
President Polk's Inaugural Address
http://avalon.law.yale.edu/19th_century/polk.asp