А ну, чики-брики и в дамки!

The meaning of the phrase "А ну, чики-брики и в дамки", which Cheeki Breeki is derived from, is related to the board game of checkers - one of the favorite pastimes of Russian inmates - and is also part of a Russian nursery rhyme. The expression Cheeki Breeki alone can mean that's everything's OK. The phonetic transliteration of the original phrase to English is "Ah noo cheeki breeki i v damké".

A common misconception of the expression Cheeki Breeki says that it is a corrupted pronunciation of "shake it, break it", but the actual phrase itself has two possible and closely related translations:

"А ну, чики-брики и в дамки" may mean "One, two, I'm on top!", which in the context of Russian checkers is said when two pieces are stacked to create a "king". Simply put, this phrase expresses that one player has the upper hand over the other, or that the speaker is in control of the situation.
Another translation relies 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 ♥♥♥♥♥♥♥ head".
It also should be noted that Cheeki Breeki can have purely cathartic function, that is, to convey excitement or anticipation. The phrase can be semantically obscure even to the Bandits themselves and therefore serve no other purpose than to display emotion and threaten the enemy, much like the American expression "♥♥♥♥ yeah!".

Looking through the game files of the Bandits, they have different types of sounds that play when they try to communicate. For example, "Ah noo cheeki breeki i v damké!" is a "detour" file that plays when Bandits try to flank the player. Going back to checkers translation, which roughly means to take advantage, it allows us to interpret the phrase as "Flank from behind!"

One of the possible reason why this expression is primarily associated with the Bandits is that, according to the lore, the majority of them are composed of criminals hiding from law enforcers. Due to being in constant opposition to the law, such people often develop an internal patois - argot - is designed to be unintelligible to non-members of the criminal community. Ultimately, Cheeki Breeki is a peculiar piece of the Soviet/Post-Soviet inmate argot.

The "Bandit Radio" track is also associated with Cheeki Breeki and is written in the genre of Chanson which is quite popular among Russian criminals and lower middle/working class people alike, street-trash and the aforementioned Gopnik stereotype. Said track is based on an existing song called "Хоп, мусорок" which loosely translates as "Hey, little cop".