I've heard it's because people will throw themselves in front of your car and claim you hit them.
Well there's also that, but apparently the people don't trust the authorities.
I did some research (quick google search):
A dashcam (dashboard camera) is an onboard camera fitted at the front window of a car and used to record traffic incidents. Dashcams are widespread in Russia[1] as a form of sousveillance, additional evidence in court, and as a guard against police corruption and insurance fraud.
The Russian Highway Patrol is known throughout their land for brutality, corruption, extortion and making an income on bribes. Dash-cams won’t protect you from being extorted for cash, because your ass shouldn’t have been speeding. It will however keep you safer from drunks in uniform, false accusations and unreasonable bribe hikes.
I'd like to note that I'm not serious about claiming Russia to be a super corrupt country, i was just poking fun at cyber's comment about Russia having very little to no resemblance to the USSR.
