"He is an America, who, leaving behind him all his ancient prejudices and manners, receives new ones from the new mode of life he has embraced, the new government he obeys, and the rank he holds. He becomes an American by being received in the broad lap of our great Alma Mater. Here, individuals of all nations are melted into a new race of men, whose labors and posterity will one day cause great changes in the world."
I think that he is trying to say that America is great because the individual can be set free and that the people who have come from all over the world to live in America can develop and become special and can have a great effect on the people who stayed where they were.
A good example of this is the singer Kelly Clarkson. Her mother was of Greek descent and her father Welsh and Irish so she is a great example of "all nations" being "melted into a new race". She came from a very normal life to win the first series of American Idol. Her debut single "A Moment Like This" broke The Beatles record for the biggest leap to number one, from 52, in the history of the Billboard Chart. She then went on to break this record with her 2009 single "My Life Would Suck Without You". She is widely referred to as the best and most successful Idol Winner worldwide and has won numerous awards. This could be said to link in with the quote above which talks about an American "causing great changes in the world".
Obviously the Kelly Clarkson example is quite trivial and there are probably many more examples of this in more important fields but I think she is a great example of what De Crevecouer is talking about in the above quote.