The bird imagery has multiple functions. On the simplest level, Chopin is writing, among other things, a local color story. On this level, the birds are used to help bring the various locales to life: to give a sense of what people would see and hear there. This goes for both the bright colors and the bits of French the parrot speaks.
However, more important than this is the symbolism. The birds symbolize aspects of freedom, especially female freedom. Some birds (those seen flying in the distance) symbolize active freedom. Some, like the parrot in the cage, symbolize a denied desire for freedom. Some, like bird with the broken wing, symbolize an inability to fly freely.