The classic answer to this question was provided by Frederick Jackson Turner in the 1890s. Turner argued that the existence of the frontier helped to make American values and society more individualistic and democratic. Turner believed that people who went out to the frontier became tougher and more able to care for themselves. He held that the experience of being constantly on the move made Americans less set in their ways and more flexible. He argued that these influences created a society in which Americans were rugged individuals who saw one another as equals and who were ingenious and creative in finding ways to cope with adversity.
It is highly probable that the vastness and challenges of conquering much of the North American continent helped shape the values and culture that can be considered uniquely American. The other answer posted here references the comments of Frederick Jackson Turner, perhaps the preeminent proponent of the suggestion that, in Turner's words, "the existence of an area of free land, its continuous recession, and the advance of American settlement westward, explain American development." Others, including contemporary historians, have similarly advanced the thesis that the expansion westward of the United States helped immeasurably to shape the values and culture of the rapidly growing nation. That said, there is not a consensus that this was the case. A 1981 issue of the scholarly journal American Quarterly, for example, includes copious details intended to refute Turner's conclusions [See "The American Frontier," American Quarterly, vol. 33, No. 5, 1981].
Whether true or not, the sense that America's westward expansion, with the natural and man-made challenges that involved, including the brutal, arguably genocidal policies directed against the region's native inhabitants, has become such a part of the nation's ethos that it is probably impossible to separate fact from fiction on this subject. The United States having expanded from coast to coast, the next challenge was space, and it was common for Americans to discuss the nation's space program in terms of an extension of Manifest Destiny. A 1986 study sponsored by the National Commission on Space titled "Pioneering the Space Frontier" presents the challenge of manned space exploration in precisely those terms (hence, the report's title). Voluminous articles and studies on the space program have similarly presented American plans to explore the solar system and beyond in the context of a Manifest Destiny for space.
The notion of rugged individualism and self-sufficiency central to the psychology of Manifest Destiny is more than mythology. It has permeated American culture and instilled a sense of mission that continues, albeit under a more constrained budget. The sense of having to explore and conquer new frontiers, whether land, sea, or space, is an enduring characteristic of America, and westward expansion was only the first such manifestation of that ideal.