This question was the subject of Frederick Jackson Turner's "Frontier Thesis," one of the most famous and important interpretations of American history. Turner, writing in 1893 (three years after the US Census declared that the frontier was closed) argued that the existence of what he called "free land" in the West had contributed to the development of democracy in America by making American society more egalitarian, or equal. In Europe, poor people had been condemned, due to land shortages, to work as peasants whereas, in the United States, people just moved west and claimed their own land to work. Even after the Industrial Revolution, the fact that people had this option meant that they were less likely to form an oppressed working class. Turner saw the "Jacksonian era," the period referenced in the question as the best example of this phenomenon. Americans moved from the East coast to the emerging West (in those days the Mississippi Valley) in large numbers, seeking their own lands. These people, many of whom made their start as small farm or plantation owners, embraced a new, democratized form of politics that appealed to their interests. "Jacksonian Democracy," the name given to this new democratic political culture, saw the West as the future, and resisted such institutions as banks and high tariffs, which were seen as beneficial to Eastern financial elites. Most important, many of these new states extended the vote to all white men, a trend which emerged in the East as well. In short, the West opened new opportunities, both economic and political, to ordinary men.
Modern historians, however, strongly dispute this thesis, especially by demonstrating that this new democratization of American politics was based on limiting the freedoms available to others. Most of the so-called "free" land in the West, for instance, had been opened by the removal of Indian peoples in the period 1820-1840, and the same period witnessed the explosion of the cotton economy and the resulting spread of slavery into the region. Even if ordinary white people, the bread and butter of Jacksonian Democracy, didn't own slaves themselves, they certainly aspired to own them. Historians tend to emphasize that slavery, in many ways, was at the heart of the economic changes that made westward expansion and the spread of democracy possible.