Symbolism, Imagery, Allegory
Chekhov was totally into birds. He raised tame cranes when he lived in Yalta (purdy, huh?) and he wrote a whole play called The Seagull. This guy just couldn't get enough of his flying friends!
Birds play a big role in The Three Sisters, too. Tuzenbach uses them as a metaphor for the human journey: "Birds that migrate—cranes, for instance—just fly and fly… they keep on flying without knowing where or why. They fly, and they will always fly… they can talk philosophy if they want, but they can never stop flying" (2.82). Birds talking philosophy. Pretty deep, huh?
Meanwhile, Chebutykin calls himself "a migrating bird that's too old to fly" as he urges Irina on to happiness (4.42). And Masha, longing for escape, notices the birds "migrating already. [...] Swans. Or maybe they're geese. You happy things…" (4.74). Birds are symbols of freedom, mobility, and beauty.