The main thesis in Frederick Jackson Turner's "The Significance of the Frontier in American History" is that "The existence of an area of free land, its continuous recession, and the advance of American settlement westward, explain American development." For Turner, American history and character were largely explained through a movement westward, with successive waves of "fur-trader, miner, cattle-raiser, and farmer" into empty, undeveloped land. That is because of the American frontier.
The American frontier differed tremendously from the frontiers of Europe, and consequently, America differs tremendously from Europe. In Europe, land was densely populated, with one heavily populated nation bordering another heavily populated nation. America's western frontier was full of unclaimed, undeveloped land, and according to Turner, that made all the difference. It did so in four important ways.
First, this expansion enabled America to have a "composite nationality." It was not an English nation, a German nation, or any other kind of European nation. The country became a mix of different European races and, thus, uniquely American.
Second, the US became less dependent on England. As the frontier kept moving further west, it provided an incentive for the coastal states to develop manufacturing and trade.
Third, it strengthened the federal government, which oversaw this westward expansion.
Finally, and most importantly, it strengthened American democracy. Individualism flourished at the frontier, and it was from the west that demands to increase suffrage (i.e., the right to vote) constantly arose. Freely available land in the west also ensured that anyone could work their own patch of land and be self-supporting, not dependent on others. Economic independence enabled political independence.
Turner's thesis remains highly contested. For instance, his talk of empty, undeveloped land largely ignores the presence of Native Americans, and at times he downplays the role of slavery in American history. But it's also highly influential—if you've ever watched a Western, it probably drew upon Turner's ideas.