The historian Frederick Jackson Turner is famous for his theory about this question. He argued that it was the frontier that shaped America's culture. He argues that the fact that Americans could keep moving west (largely into the Louisiana Purchase) is what made American culture what it was in his day (and continues to some extent to be today).
To Turner, the frontier made Americans stronger, more individualistic, and more adaptable. When settlers moved out into the frontiers, they could not rely on others or on the government. Instead, they had to be strong enough to take care of themselves and they had to be individualistic enough to be able to rely on themselves. They also had to be inventive and adaptable enough to deal with whatever happened to them.
We in America like to think that these are traits that characterize our culture. If we are right and if Turner was right, westward migration shaped our culture in a way that allowed us to become strong and self-reliant