Diseases often have trouble jumping from one species to another. The closer related to you your food is (evolution wise) the smaller the gap the disease needs to jump. This is why eating bushmeat from apes is more risky than eating some less closely related animals.
The above is true especially for viruses. Parasites are another problem. One thing that has been big in the news are prions.
Prions aren't really living things at all. They are just certain types of proteins which cause the proteins inside of you to become like them by their mere presence. In particular the prions everyone is worried about are ones that are inside your brain and nerves. If you eat another humans brain who has such a prion disease you will become infected yourself.
Normally this is not such a big deal as people aren't in the habit of eating each others brains.
However in cultures where cannibalism was practiced, such disease can become a problem. One example is Kuru also called the laughing diseases which used to be found among cannibals in places like New Guinea.
Another example is mad cow diseases which became a problem when stupid people got into the habit of feeding cows the ground down remains (including brains) of other cows.
In fact mad cow became such a big problem that enough cows with it ended up in the food chain to possibly infect humans despite the species gap. Creutzfel-Jacobs is what the disease is called that many think is what happened if you eat cannibalistic cow brains.
The TL;DR of all this is that if you plan to eat humans make sure to only pick healthy looking ones, stay away from their brains and avoid eating any known cannibalis who seem to have the shakes.
Many diseases are limited to a single species. So, by eating other kinds of meat there are a lot of infections that may be in it, but just isn't an issue anyway. While, by eating meat of the same species you are receptive to all the diseases in the meat.
The extreme case are prion diseases, they are misfolded proteins that when in contact with the same protein can cause it too to become misfolded. Again, since the same species contain the same proteins, the risk is much greater. These diseases are really nasty btw, often neurodegenerative.
Creutzfeld - Jacob's (mad cow disease) is a prion disease you can get from eating your own kind. But that's about all I know about it.
Edit: eating your own kind
In general...it's not. Eating properly cooked 'meat' of the long pig isn't that much more risky than eating other properly cooked meat. The exceptions are if the meat is diseased, since that disease already works well with humans. Diseases that attack animals are specifically formulated for that animal, and it is much more difficult for them to 'jump' to humans. They usually have to mutate to some extent. No need when it is human to human.
And the biggest exception is eating human brains. Cannibals should stay away from that, because nerve and brain problems jump even more easily. Certain protein chains (so not bacteria or anything) in a bad brain can replicate in the eaters brain and cause the same problems.
As for other species, it takes time for the protein - brain problems to show up (the protein needs to make lots and lots of copies). For animals with short lifespans, there's just not enough time for it to make a difference. If it takes 10 years for the disease to really kick in, and an animal can make 40 babies in that time frame, evolution doesn't really have an issue with it.
Cannibalism used to be(still is for several tribes) a huge part of early, native society. You can actually eat plenty of the human body, but several areas(specifically muscles) house neurotoxins which can do all types of damage to the body. This is why we have stories like the wendigo(which prompted the term wendigo psychosis) that tribes would recant to warn future generations of the harms of cannibalism.
You can read about the Algonquian myth here, just the best example off the top of my head: https://en.m.wikipedia.org/wiki/Wendigo
Thinking of partaking in the other other white meat are we?
The biggest danger with cannibalism is prion disease. Our bodies are made up of proteins that "fold" to form a certain structure. It's a very complex topic but basically proteins need to be a certain shape in order for them to perform certain tasks in our body. Proteins sometimes don't fold correctly though. These junk proteins, called prions, can do wacky things, and can result in cell death. If they infect they brain, they can cause sponge like holes to form. This is essentially what Mad Cow Disease and it's human equivalent Creutzfeldt–Jakob Disease (CJD) are. They have similar symptoms to dementia and Parkinsons as the brain starts to waste away. The disease is fatal.
If you ingest the brain or other nerve tissue of someone infected with CJD, those prions enter your body and start to attack your nerve tissue as well. A condition called Kuru. It was first noted in the Fore people of Papua New Guinea. The Fore practice a form of ritualistic funerary cannibalism.