The guess here is that the Celtics were comfortable with their big-man depth with Jared Sullinger making a surprise return last week. That made Randolph, who played only 25 total minutes in five appearances since being acquired from the Phoenix Suns in mid-January, an expendable player, particularly as an expiring minimum salary for a team comfortably under the luxury tax line.
The Celtics did not disclose details of Babb's contract, but have used chunks of the mid-level exception in recent years to sign young players to nonguaranteed deals (see also: Chris Johnson). This puts the player under Boston's control moving forward, while also making them trade assets after the season. The Celtics could spend up to the remaining level of the midlevel exception to sign Babb.
It would be another asset for Celtics president of basketball operations Danny Ainge and his staff to maneuver with as Boston rebuilds its roster.
Babb was immediately assigned to the Maine Red Claws of the D-League, which means virtually no change in his situation. He'll stick with the Red Claws for the upcoming D-League playoffs.
If Boston is still active when Maine finishes its playoff run, both Babb and rookie James Young, currently on assignment with the Red Claws, can rejoin the Celtics for the remainder of the season. Barring injury, it would seem unlikely Babb would be active. He was signed for emergency depth at the end of last season when Gerald Wallace required season-ending knee surgery, but was waived in late September due to a roster crunch.
[Update: Babb's deal appears to be a two-year, minimum salary signing. He'll earn roughly $48,000 for being part of Boston's roster for the final 11 days and will have a nonguaranteed salary of $947,276 for next season.]