A doctor weighs in . . .
Frankenmauser's got it right, for the most part.
The lungs are elastic, and want to collapse. The chest wall and diaphragm want to expand. They are held together by the vacuum between them. Punching a hole in the chest wall allows air in between lungs and chest wall. Now the lungs collapse, and there is no seal for the diaphragm and chest wall to pull them open. No breathing, no oxygenation of blood or elimination of carbon dioxide. Unconsciousness in 1-2 minutes at rest, but faster because the deer is trying to run away.
The lungs are also extremely vascular, as might be expected. Punching a hole in them allows bleeding into the chest cavity, which is freshly vacated by collapsed lungs, as well as into the airway (and out the mouth and nose). The closer to the central vessels the shot hits, the faster the bleeding. The faster the bleeding, the less blood makes it back to the heart to be sent to the body, and the faster blood pressure and the animal drop.
So a good double lung shot causes catastrophic 2 organ system failure, a quick demise, and no lost meat.