"Manifest Destiny" is a term the American Government invented to push the drive of settlement to the West. The American Government used propaganda to encourage settlement- it was a general belief that Americans were god's people destined to be great and fill the continent from East to West. Advancements in technology helped push West as well, with the steamboat, telegraph and railroads bringing new links for commerce and trade in the East.
It's important to remember that America had begun intense trading with other countries where markets had just opened (the slave trade and Asia are two notable examples) and demands for goods and commodities in the East was rising.
Americans had a strong patriotic belief in their country. They ignored the Plains Indians, as Americans believed them to be savage pagans.
The reason for the propaganda and push West was that land acquired through war or through sale (Louisiana Purchase) was in danger of being taken or filled by other countries (in particular Great Britain and Mexico). Problems in the East such as overcrowding and unemployment were other factors in the push West.
Here is a list of some of the settlers who fulfilled the Manifest Destiny:
The belief in Manifest Destiny had some role in the Mexican War. The belief that Americans were the people fit for the land, meant that when rumors spread that Mexico and Great Britain wanted to seize control of California, there was a change in the nature of the belief. It was more militant and aggressive. The war with Mexico occurred because of this belief. James Polk pushed the war as he believed the (false) rumors that Mexico and Great Britain were collaborating to interfere with the U.S. expansion. This also lead to him annexing Texas.