While there is much to be said for the previous answer, there are also places where I would take issue with it. Mainly, it is very hard to agree with the idea that those who won the war were “not landed wealth, established gentry, and did not come from entrenched interests of power.” If we look at the people who came to power after the Revolution, it is hard to say they were not wealthy members of the gentry. Washington and Jefferson were wealthy plantation owners. John Adams was from a prominent family. Thus, the major leaders who came out of the Revolution were the same people who had been in power (at least among the colonists) before the Revolution.
To talk about the way in which the Revolution was radical, I would turn to Gordon Wood’s The Radicalism of the American Revolution. Wood argues that the true radicalism of the Revolution came from the idea that it inspired the lower classes. He says that the leaders of the Revolution expected that they would be the leaders of the new country. They anticipated that the men of leisure in the new country would govern on behalf of the common people. Wood says that the rhetoric of the Revolution helped to make this impossible.
Instead, Wood says, people came to believe that society should not be organized vertically. There should not be any idea that some people were better than others. What the Revolution did was to ensure that the lower classes felt that they were just as good as the people like Washington and Jefferson. In this view, the radicalism did not really show itself completely until the US began to democratize in the early 1800s.
According to Wood, then, the radicalism of the Revolution was not about who won the Revolution but about the vision of society that the Revolution’s rhetoric imparted to the common people of the new country.
If "radical" can be seen as groundbreaking or something unprecedented, I would suggest that the American Revolution represented this spirit in the manner in which the war was won. One reason would be that the victors of the American Revolution were unlike anything seen before. In a geopolitical setting where power rested with the established power structure in Europe, its rejects in America asserted their own power against one of the mightiest forces in the world. The American Revolution could be seen as radical because the people who won the war were not landed wealth, established gentry, and did not come from entrenched interests of power.
The victors in the American Revolution were more "regular" than any established power in the world. The narrative of the leaders such as Ben Franklin or George Washington was fundamentally different than someone like Cornwallis, King George, or even Braddock. The new leaders of America were fundamentally different than anyone assuming leadership in a nation. Washington himself noted the radical nature of the outcome of the Revolution:
[The events that led to American victory] have seldom if ever before taken place on the stage of human action, nor can they probably ever happen again.
Washington understood the unique nature of the victory in the American Revolution. He understood that regular people, people from modest means, were able to launch a successful offensive against the strongest nation in the world. Washington recognized that the government that would replace Britain would be one consisted of people who were more modest and more reflective of "common folk." The new government would be one that could reach closer to the ideal that Lincoln would outline a century later in being a government, "of the people, for the people, and by the people." For this reason, the outcome of the American Revolution can be seen as radical.