As a child Lev was a brilliant chess player, but wasn't able to reach the "higher plane" that would've allowed him to continue to be brilliant as an adult. Since chess is a strategy game, it works throughout City of Thieves as a metaphor, as Lev and Kolya discuss their plans and strategies for staying alive as well as finding eggs. It also serves to differentiate between Lev, who is described as being small and weak, and other more conventionally masculine and strong characters like Kolya, the giant, or Abendroth. While Lev is certainly at a disadvantage for most of the novel because of his fear and slight build, it's his knowledge of chess that allows him to strategically position himself, Kolya, and Vika in a place to both beat Abendroth at chess and kill him.
Chess Quotes in City of Thieves
"We're pawns and he's a rook, that's what you're saying."
"We're less than pawns. Pawns have value."
"If we can take a rook, we have value, too."