I don't really care to get into a long treatise on why that is a facile understanding of the term. But in short, it's really an idea used by social and political philosophers to explain aspects of America's development as a nation that are different than, say, aspects of the development of European states. It doesn't mean better, it mean different.
Professor Louis Hartz, for example, emphasized the lack of a feudal tradition in America. He argued that this helped to explain the lack of significant left/socialist tendencies on the one hand and aristocratic tendencies on the other, which, it's been suggested, helps to explain the origins of a sort of centrist, liberal consensus in America.
Like I said, I'm not trying to get into it, only to say that it's a theory, or a set of theories, that explains difference. I'm not denying, however, that some writers have interpreted the term to suggest superiority, only that it would be giving too much credit to the current mis-users of "American exceptionalism" to believe that this is the origin of their interpretation.
The problem for me is when progressives start throwing the idea around. It's like they feel they have to do conservatives one better by agreeing that great things are only possible in the USA and other countries are second best by the simple fact of not being America.
You may have seen Chris Matthews on MSNBC doing a little segment about how Obama once commented that his story would not be possible in any other country and that America was exceptional in this way.
I can only say that for anyone who believes this, you really should get out more.
America has had more than its fair share of challenges as a leader in securing racial equality for its people and Obama's rise to the presidency should be seen as the extraordinary event we all know it to be, not an opportunity for blind self-congratulations. Obama was and is swimming against the current, not with it.
Anyway, if you haven't seen the clip to which I refer, here it is: