This is called regurgitating. Birds regurgitate food from an organ called the crop. This is NOT vomiting. A crop is NOT the same as a stomach, but it is a digestive organ - food is eaten and goes into the crop. From there is passes to other organs in the digestive process.
Birds regurgitate to feed their mate, or to feed their chicks.
It is a natural part of parrot behavior - as far as I know, all parrots do this, not just eclectus. As it will try to 'bite' my finger, i always find there is food in its mouth.