What exactly does cooking meat do as far as sanitation goes?

Very simple: it kills bacteria.

The recommended cooking temperatures are empirically derived. They lead to a log 7 reduction in the amount of bacteria present. That is when you have a piece of meat, 1 in 10 000 000 of the bacteria on it remain alive after cooking. The recommended temperature for storage and cooking is chosen in such a way that, for food which has been handled safely before cooking, this reduction is enough to produce food which will be safe for the next 3-5 days in the fridge.

There is no path from "unsafe food" to "safe food". There is no telling how many bacteria are on your food if it was not handled properly. Bacteria grow exponentially, so small differences can result in not just a few more bacteria, but in many times as more bacteria. Also, there is always the problem of toxins created while there was a lively bacterial colony on the meat, even if you cook the bacteria to death afterwards.

In short, food is only safe when it is handled safely before cooking, cooked safely, and handled safely after cooking. A slip at any step cannot be remedied.

I'm no expert so don't risk your health based on what I say. There are two threats in raw or spoiled food - pathogens (bacteria and fungi mostly) and the toxins they produce. Cooking food kills pathogens but usually cannot destroy toxins. So cooking does not make spoiled food safe because the toxins are already there and it's too late.

If the food hasn't spoiled or expired yet, and so there aren't a lot of toxins built up yet, you still need to kill the pathogens so they can't hurt you after you eat them. High temps will kill them. As an interesting note, the recommended temperatures are what is required to kill bacteria instantly. They will still die at lower temperatures, but slowly. You can plot a graph of temperature vs time to make things safe to eat. For chicken, for example, 165 will kill everything in just a few seconds. But you can actually hold it at 160 or 155 for long enough to make it safe as well. I once ate sous vide chicken that was cooked at 145 and it was fine, but you need to hold that temperature for about 90 minutes to make it safe.

Remember that cooking doesn't completely sterilize anything. If you leave cooked meat out at room temperature for more than 2 hours, you're risking the bacteria recolonizing it and making it unsafe again. I did recently eat some turkey chili that I accidentally left out on the counter overnight and I was fine (I boiled the hell out of it first) but most people would advise against that.

Also remember ground meat needs to be cooked through completely. Most bacteria live on the surface, so searing a steak is usually enough to make it safe, but grinding mixes the bacteria into it so it needs a high temp all the way through.