This term has come up mainly in conservative channels and it was summed by physician and GOP Presidential candidate Ben Carson. Paraphrasing he noted how when America arrived on the scene the world was stuck in doing things the same way. Then 200 years after America arrived on the scene humans walked on the moon....
Perhaps what Ben Carson is driving at is how in a short time Americans became successful in a variety of endeavors. We were determined to beat the Soviet Union to the moon during the 1960s. We made the technology to get humans there.
I've learned over the years that Americans are open to new ideas especially ideas that can serve to change our lives. In the last 100 years we have electricity, light bulbs, aircraft, automobiles, telephones, etc and they're commercially available for everyone to either own or use today. Today we're still innovating and finding things to improve upon.
America isn't perfect, no nation created by a person or persons truly is. People can debate all day whether on not this is the home of the free, but we are free in many ways. More free than people in other countries.
I'm not sure I've totally explained American exceptionalism, but I would like to say that our products aren't what we spread to the world. In fact what Americans can spread around the world is the philosophy of individual liberty. It's a small part of what it means to be an American, but without recognizing the liberty of an individual nothing else really follows.