That was based on the reports he got from CBS affiliate KRLD reporter Eddie Barker. Barker was the first to say that the President was dead.
Wikipedia indicates that Rather supposedly called the priest who gave last rights. No citation or indication how Dan Rather would know who the priest was or how to contact him directly.
I seem to recall an interview from the 1980s where Rather admitted he had a hunch that JFK had passed and ran with it.