These birds have an unbelievable sense of smell along with a very strong immune system to fight bacteria found in decaying carcasses. If ingested, even in much lesser quantities than what these guys are consuming in one single meal, such bacteria would easily kill animals or humans.
In the frame below, the vulture struggles with the weight of the carcass:
Trying to get a better handle on the carcass using its talons:
Lastly, a better grip holding the carcass down with both its feet, allows for easier feeding: