Why does the bird beat its wings sympathy

Answer:It really, really wants to leave that cage, so much so that it keeps hurting itself (eesh). Each time the bird beats its wings against the bars, its old scars "pulse," or throb again. This suggests the idea of recurring pain. The bird suffers over and over again; its pain isn't limited.

Source

Sympathy Summary | Shmoop

the bird still sings inside the cage, but it is not the same. The song thus represents anything that a creature is born to do; in the case of the speaker, he is born to write. If he is free, his writing will represent a more contented state of mind than if he feels trapped