Well, it really wasn't a decade or two. America had rich democratic traditions going back not just to the colonial period, but to the history of England itself. In England, the power of the King was limited ever since Magna Carta in the 13th century. Over the centuries, the Parliament of England gained more and more power vis-a-vis the King. English (and then British) subjects (which included American colonists) got to enjoy considerable rights which were enshrined into law. American colonies were, to a considerable extent, essentially democratic, with legislative assemblies making the laws, and only minimal interference from Britain. There already was a fairly high rate of political awareness and participation in the political life of the colonies among citizens. All the fundamental institutions necessary for a successful democracy - freedoms of speech and press, property rights, a functioning and independent judicial system and so on. Moreover, culturally the people were accustomed to enjoying certain individual and political rights. In fact, those rights being threatened - the Parliament taxing the colonies without their consent - is what sparked the American Revolution in the first place. So in short, the United States didn't really need to create democracy from scratch. All that needed to be done was sever ties with Great Britain, and replace them with a federal system.

Some other countries have different cultures. Liberal (in the classical sense) values aren't as deeply rooted there. Institutions necessary for democracy to thrive are not as strong (or maybe not existent). There may be deep rooted problems in their society such as excessive corruption, poor legal infrastructure and enforcement, a lack in the rule of law, a high degree of cronyism/kleptocracy etc.