It means:
Ahnuu, cheeki breeki i v damke!
(1, 2, you’re on top!)
“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 fcking 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 “Fck yeah!”.