The "problem" with raw meat (for lack of a better word) is that there's a chance that it's contaminated with bacteria that can make you sick.

A very common bacteria found in raw meats (all meats: beef, chicken, pork and so on) is Campylobacter jejuni. It's a natural colonizer of the guts of many animals. When an animal is slaughtered, the contents of the animal's digestive tract should remain contained and shouldn't come into contact with what will become the meat, but this is not always possible. It can happen that meat can become contaminated with C. jejuni (and other contaminants) even though all the normal precautions have been taken.

If your food is contaminated with C. jejuni and you cook it properly, you'll kill the bacteria and make the food safe to eat. But if you eat the live bacteria, you can end up with an infection of your gut called campylobacteriosis. It won't kill you (unless you were already very sick to start with), but it'll make you really hate your life for about a week.

A lot of things have to go wrong all at the same time for you to end up eating live C. jejuni bacteria. But it does happen, to the tune of about two million times a year in the United States alone. Properly cooking your food entirely removes the possibility of infection by C. jejuni, but of course we don't always want to cook our food to the point where bacteria die. So we accept the risk of getting food poisoning. But the trick is to understand that risk so we can minimize it.

All that to say that yes, there certainly is a problem with raw meat, but it's a problem people can choose to live with if they want to.