Variant Creutzfeldt Jakob disease (vCJD) is the human equivalent of mad cow disease (Bovine Spongiform Encephalopathy, or BSE). It's caused by a misfolded form of a specific protein called a prion that can, in turn, trigger normal forms of the protein to abnormally fold in the same way. Nerve cells carrying this abnormal form are prone to death, resulting in a deterioration of mental capacity in the victims of CJD. The diseases is thought to be transmitted when individuals come in contact with infected nervous tissue. Thus, careful butchering of animals to avoid any brain or spinal materials greatly reduces the danger of transmission, and was thought to be protective.
So far, the US hasn't had to worry about significant BSE, although it has snuck across the border from Canada, where it appears to be slightly more prevalent. But it is a worry in non-domestic animals here, primarily deer and elk, where it's called chronic wasting disease. Since deer and elk are hunted extensively, the hunters have reasons to educate themselves. The advice given there is focused on avoiding exposure to nervous tissue as well.
The thing that's always confused me a bit about this advice is that the vertebrate body is pretty much permeated by nervous tissue. Most meat consumed is muscle, and the state of all muscles has to be controlled and sensed by the nervous system. The nerves that do so extend right back to the spinal cord, and so are in indirect contact with the brain. So, it's not surprising to me that a study published in Science has revealed that mice engineered to express the appropriate gene from deer can pick up prion disease from their muscle.
In a short write up in the New York Times, it's noted that nobody's actually caught vCJD from a deer yet. Hopefully, the recommended precautions have been and will continue to be sufficient. But CJD has a fairly long incubation time, and chronic wasting is still spreading across the US. If the mouse experiments do accurately suggest a route for transmission, that "yet" maybe be a significant qualifier.