The meaning of the phrase "А ну, чики-брики и в дамки" which Cheeki Breeki is derived from relates to
Chess Checkers and is also part of a Russian nursery rhyme.
The misconception of the expression "cheeki breeki" says that it is a corrupted pronounciation of "shake it, break it", but the actual phrase itself has two varying translations, which both have a similar meaning
А ну, чики-брики и в дамки = "One Two, then you're on top!" which is meaning to flank, or gain the advantage against the player.
Another translation depends on the Bandits heavy use of localized Russian slang. "чики-брики" is often warped into a vulgar threat of shooting someone, while "и в дамки" means "The Head" which would roughly translate the phrase into "Shoot him in the fucking head".
The "Bandit Radio" track is also associated with Cheeki Breeki and is written in the genre of "Chanson" which is a genre listened to by Russian criminals, street-trash and the aforementioned Gopnik stereotype. Said track is based on an existing track called "Хоп, мусорок" which loosely translates as "Hey-ho, little cop"