I just looked up the section of the Babylonian Talmud (Hullin 33a) cited by Tzemach Atlas in his post. (Please see my post immediately below this.) It is referring to removing a small section of flesh from the throat immediately after shechita, soaking it, salting it, and then eating it only after the animal has died.
Why was this done?
For a medical reason. This flesh was considered to be a tonic, probably a way of consuming part of a living animal without violating the Halakha of eiver min ha hai.
(Flesh and blood from living – or mortally wounded – animals was commonly thought of in the ancient world as containing the animal's life force. Its consumption was thought to transfer some of the animal's strength to the eater.)
As the Talmud says, if one wanted to remain in good health, one would do this. As with all medical advice given in ancient sources, this no longer applies (unless it has been confirmed by modern medicine). Further, it is important to keep in mind that this was not done because the meat tasted good, was a delicacy, or for financial reasons. It was done for medical reasons only and, as such, no financial or non-medical lesson should be learned from it.