Bread goes stale faster in the fridge because staling is not mainly about drying out. It is the starch inside the bread slowly changing back into a firm, crystal-like structure, and that happens fastest at cool fridge temperatures. A loaf kept on the counter or in the freezer stays soft for longer than one kept in the fridge.
When dough bakes, the starch soaks up water, swells and softens. That is what gives fresh bread its tender crumb. As the bread cools and sits, the starch molecules start lining up again and squeezing water out. Food scientists call this retrogradation. The crumb turns firm and crumbly even if very little water has left the loaf.
At the same time, moisture moves from the soft middle out to the crust. That is why a crisp crust turns leathery after a day in a plastic bag.
Retrogradation runs fastest at temperatures just above freezing, which is exactly where a fridge sits. Below freezing it almost stops, because the water is locked up as ice. So the fridge is the worst place for bread you plan to eat soon, and the freezer is the best place for bread you want to keep.
Heat melts those starch crystals again. A few minutes in a hot oven brings back much of the softness of a stale loaf. It only works once or twice, though, because each time the bread loses a little more water.
Breads with more fat or sugar, like brioche, usually stay soft longer, because fat gets in the way of the starch lining back up.
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