There are many ways for food to go stale or spoil. Some foods go stale because they undergo a chemical change. For example, the starch in bread loses its water, making the bread leathery and hard. Foods can also absorb odors from the surroundings, which is why some people don't like to refrigerate breads or cakes (especially not uncovered). Odor exchange can go both ways: some compounds from the food itself can evaporate into the air and be lost, changing the flavor of the food. Foods with a lot of fats or oils, including things like peanut butter and most bread, will go rancid as the fats or oils undergo oxidation and hydrolysis. Very sugary foods like orange juice can undergo fermentation, when yeast converts some of the sugar to alcohols. Finally, even small amounts of mold and bacteria growth can drastically affect the flavor and texture of foods.

That depends on the food in question, but for the most part it's drying out. Most foods include quite a bit of water, and as that water evaporates, the food becomes stale. This is why food gets stale faster out in the open than in a closed compartment, because the closed compartment traps the water vapor inside.