The main way in which the idea of liberty transformed American society was in the fact that it created a society that valued freedom and the rule of law.
Before the Revolution, the rule of law was not such an important concept. Of course, Britain had laws, but its legal system was not so attuned to the idea of protecting the rights of individuals and there was little adherence to the principle of limited government.
The Revolution's idea of liberty pushed American towards a society where these things were much more important. It created a society where, for example, things like the Alien and Sedition Acts would not be tolerated for long and where rights like that of a jury trial and that against warrantless searches and seizures would soon be taken for granted.