I dont know if this is so true in the United States. If it were we could eat raw bacon and undercooked meat without fear of getting any sort of food-born illness, where in someplace such as Germany they dont have these problems because they dont have mega factory farms and processing centers like we do.
I said very clean, not sterile. It is pretty unreasonable to expect any sort of food to be entirely free from all sorts of bacteria and such without it being sterilized first.
Anyway, the law in most parts of the world, including America is that slaughter houses HAVE to follow very strict rules when it comes to slaughtering animals. People watch the process from start to finish to make sure it is "humane", and pretty much every single part of the animal is tested before any part of it is put to use. The carcasses have to be displayed in a way that makes them easy to inspect (cow heads have to be displayed in a way that makes it possible to see into the back of the throat for example) and samples of all parts of the animal (meat, bone, skin, blood, intestines, stomach and so on) are taken and tested before any part of the animal can actually be used.
Even in large factories all of these have to be followed. Any slaughter house that doesn't follow these laws is like I said probably some sort of illegal operation and chances are you'll never eat meat from a place like this anyway. That or the slaughter house is in some less developed country.
The videos you might have seen of animal cruelty, blood stained buildings littered with gore and whatever else are nothing at all like your average slaughter house. (It might seem strange, but in your average slaughter house it is pretty rare to even see blood).