You can read all about it at GGW. Here's my thoughts about it.
The DC code allowing cyclist on the sidewalk goes back to DC's original traffic regulations from 1972 and 1974 and has it's roots in the Uniform Vehicle Code, which in 1968 added a stipulation that cars were not allowed on the sidewalk. I'm not sure what DC's traffic laws were like before then, but even back then it included the ban within the CBD. The primary reason for the CBD ban, as I understand it, was to deal with the problem of bike messengers on the sidewalks. The ban on sidewalk cycling (if back issues of Dr. Gridlock are any indication) was not enforced and was never successful, even when messengers were required to have license plates on their bikes. The problem was such that in the late 1980's Ward 6 Councilmember Nadine Winters proposed regulations for the bike messengers, including requiring them to be licensed, pass a safety course and wear visible ID numbers. But with the decline of bike messengers in DC, the perceived need for such regulations has largely disappeared. But the CBD sidewalk ban remains.