It's basically a simplified version of the Great Seal of the United States. It was made official insignia in 1832, source. The source has more about US rank insignia development (cockades in caps and all that).
There must have been a reason though, it must have been in informal use.
Wikipedia thinks this that it was used in 1805, but it's unsourced, my searches only lead to people verbatum quoting wikipedia and it got the other date wrong so I think we can disregard it for now.
Wikipedia has another "fact", backed up by Ask (because of plagiarism): "There is evidence that colonels wore the eagle as rank insignia in 1829 when they transferred the gold or gilt eagles that decorated their hat cockades to their collars". Here the trail leads back to this Office of History (Sacramento Air Logistics Center, McClellan Air Force Base etc) document, which has more info unfortunately doesn't elaborate further about origins.
So I think that it came into informal use and was adopted. I don't know why colonels specifically were able to do this, perhaps their uniform was ill defined or maybe it was just luck. But it is certainly not surprising that the eagle, as such a powerful symbol, made it's way in somewhere.
The eagle doesn't appear similarly in the british or french rank system (obviously it is a powerful military significance to the french, but not especially to colonels), which the two systems it's reasonable to assume the American rank structure emerged from. So it's origins are not there.
The great seal of the US wikipedia page will explain it's significance to the American state and by extension it's significance to the military. Which is background, but doesn't address the colonel question.