Hi there. Well actually, the person who suggested popping the sac with a sterile needle is correct- so maybe they are more of a bird expert than they think!!
The reason for the swelling is something called a ruptured air sac- birds have many airsacs (they do not have lungs like ours- they are connected to air sacs), these air sacs are filled with air so that the bird is light and can fly. Sometimes these sacs can rupture, and when they do the air builds up under the skin.
The treatment is to let the air out via a small nick. This may need to be done repeatedly, making sure you use sterile precautions( sterilize the skin to be punctured with surgical spirit or povidone iodine from the pharmacy).
If the air sac keeps leaking, the swelling will keep refilling- the only solution in this case is for a vet to make a permanent hole to let the air out of.
Andrea
Bird Veterinarian
15 years in companion animal practise, BVM&S degree Edinburgh university