Good questions.
Different sources ...
September 2015: FBI Informs DNC of Russian Hacking - The Moscow Project - NYT
The Perfect Weapon: How Russian Cyberpower Invaded the U.S. - NYT
Looks like FBI awareness may have arisen from reports to it, from others. Phishing email, perhaps with "To" headers that included DNC, or maybe the intrusion was used to send further phishing spam, this originating from DNC servers. You've likely received e-mail from a friend whose system was infected.
Yared Tamene, the tech-support contractor at the D.N.C. who fielded the call ... "I had no way of differentiating the call I just received from a prank call," Mr. Tamene wrote in an internal memo, obtained by The New York Times, that detailed his contact with the F.B.I.