Washington explained it this way: “We take the stars from heaven, the red from our mother country, separating it by white stripes, thus showing that we have separated from her, and the white stripes shall go down to posterity representing liberty.”
http://www.va.gov/opa/publications/celebrate/flag.pdf
Are there any sources that support this quote?
Edit: I forgot to mention the closest thing to a source I could find is this Congressional Record book
I doubt it. A few quick searches of Founders Online (a database which includes the text of all the 62 volumes of the Papers of George Washington doesn't turn up anything that seems similar to that quote.
I think this is a result of "academic inbreeding"--meaning, someone, somewhere said a thing and it got published in a book, and an influential book repeats the thing and cites that book, and then lots of later books cite the influential book and repeat the thing, and as a result everyone thinks the thing is true because it's repeated in so many different publications when in reality it was just the one guy who said the thing without anything solid to back it up.
As far as I can tell, the earliest instance of this quote comes from Peleg D. Harrison's The Stars and Stripes and Other American Flags published in 1905. Here's a copy of the quote from the 5th edition that book (on p. 59).