How did the US manage to expand its territory across the continent

With regards to asking the land from the Native Americans, they promised them land, and made them sign contracts that would entitle them to their land back, and they still continued to claim their land, regardless of this fact.

Manifest Destiny. Many politicians got Americans to believe that it was their destiny to span the continent and claim all the land. This even was also know as Westward Expansion, and refers to the time period where Americans migrated west for a new life or economic pursuit.