The Louisiana Purchase helped to create a uniquely American culture by drastically increasing the amount of "frontier" for Americans to expand into. This helped to create a culture that was fiercely independent and individualistic. It also helped to move America towards more democracy.
By acquiring such a huge amount of land, the Louisiana Purchase gave Americans a great deal of space into which to expand. It allowed people to go out into the wilderness and fend for themselves instead of staying in more populated areas and becoming urbanized. This helped create America's frontier culture.
This also helped to make America more democratic, as was manifested in the Jacksonian Era. The large number of people who were living out in frontier areas helped to influence America to become a country that rejected elites and gave power to the "common man."