Like Ancient Romans in 20 BC, Americans had a belief in the 1850s and onward that their nation was destined (obviously) to expand as a great power from ocean to ocean and rule everything in between. The destiny is to rule and the manifest means obvious. For Ancient Romans this meant conquering far off lands like the Netherlands and South Germany and Switzerland. It even included Britain (that had to wait 60 years) and the Parthian Empire, centered in Iraq (that did ntoi really happen, diplomacy solved the matter). For Americans in 1850s-1890s it means all the territory not yet organized into states from the Mississippi west to CA, Oregon, Washington. Other nations also claimed the land from Texas to Washington and everything in between - often two or three other powers from Great Britain, Russia, Spain, later Mexico, and briefly France. And most of it was inhabited by native peoples (Indians). The impressive result saw mostly private citizens benefitting from generous government policy settle that land in the frontier tradition far more widely and successfully than the other powers could. The US got 90% of the land, maybe more. The only part the US did not get was British Columbia, which we claimed but yielded (and never tried to settle). The US did not seriously claim Baja California, which has always been part of Mexico.
The use of the term makes the claim that the US is divinely favored to control the land for the bettering of the world. American government policy allowed people to go with little regulation to settle the West "out on the frontier" any way they wanted. This created American "rugged individualism" and gave rise to the belief in "American exceptionalism."