It's not a question of "in the old days", it's a modern practice for half the world's population. Many countries lack a refrigerated chain from abattoir to grocery, where open air butcher shops are common. Freshly butchered meat is stacked or hung in the open air, pretty much all day in hot climates. Sometimes for two days. Here's a typical butcher's shop in Pakistan. You'll find thousands of these across south and southeast Asia, China, and other parts of the world.
It was quite fashionable in Europe to hang game birds in the cellar to tenderize the meat and improve flavor. It's the same principle as dry-aged beef that you can buy from your grocery store today. And while cellars were colder than the temperature above ground, they were nowhere near refrigerator temperatures. It was typical to hang pheasants and partridges at 55 degrees Fahrenheit for about 2 weeks at the Savoy Grill in London, for example. People paid premium prices for such poultry.
If you're wondering "how do people survive", the answer is quite simple - cooking. All meat at the butcher has bacteria growing on it, even the kind you buy plastic-wrapped package in the cold section of your local grocery. If you keep the temperature low enough (below 40 F), bacterial growth is slow and the meat lasts longer. If the temperature is high, bacteria grow faster. The difference isn't one of bacteria or no bacteria, but simply of bacterial load.
Cooking reduces the bacterial load by killing bacteria. Cooking also destroys most toxins produced by bacteria, with a few notable exceptions, like botulinum toxin produced by various species of Clostridia. Luckily, these bacteria require anaerobic conditions (low oxygen) to survive, so you won't find them in meat that's left in the open air.
Generally, meat is cooked quite thoroughly in areas where such open air butcher shops are found. You'll find a lot of meat stews, high heat tandoor ovens. Not so many semi-raw steaks or burgers. This makes the meat safe to eat.
Of course, food borne illnesses do happen, if the meat is cooked improperly. You'll often come across reports like "dozens of guests taken sick at wedding dinner" in local news. This happens because the food is cooked on site, in temporary kitchens specifically for the occasion, and mistakes happen when you're cooking for hundreds of guests in a hurry, on makeshift premises.
But by and large, cooking at the right temperature for a sufficient period of time renders most food (including meat) safe to eat. As I said, there are some exceptions such as botulinus, but these are more a consequence of canning techniques which create anaerobic environments, so they are more a "modern" thing.
Cooking can also be used to prolong the shelf life of already cooked food. For example, in places without refrigeration, people will often "re-cook" the day's leftovers to use the next day. This simply involves bringing the food to a boil, then cover the pot and turn off the heat. A quick form of pasteurization that resets the day's bacterial load back to zero, giving the food a few more hours of safety.
It's also worth noting that "spoilage" doesn't equal pathogenicity. Spoilage is simply a change in the food's appearance, taste or odor, which people find unpleasant. It can be caused by many bacteria which don't cause disease in humans. Pathogenicity is due to bacteria that are known to specifically cause disease. Sometimes, the two can work at opposite ends, for example lactobacillus (which causes spoilage) is itself harmless to humans, and even inhibits the growth of some pathogenic bacteria.
This is more a topic in biology and food science rather than history, so if you care to read more, I can point you to some resources:
GILL, C. O., & NEWTON, K. G. (1980). Growth of Bacteria on Meat at Room Temperatures. Journal of Applied Bacteriology, 49(2), 315–323.
CHUNG, K, DICKSON, J & GROUSE J. (1989). Attachment and Proliferation of Bacteria on Meat. Journal of Food Protection, 52(3), 173-177.
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