Dehydrating is the process of losing moisture. It happens as free water inside your bagel slowly finds its way to the surface and leaps off into the atmosphere, never to return. In the open, this happens pretty fast, but you can mitigate it by storing bagels in a plastic bag to prevent excess moisture loss. This will, of course, soften up the crust, but we’ll deal with that later.
Staling is the retrogradation of gelatinized starch into a rigid crystalline structure. In other words, when bread is fresh, its starch is fluid. It moves around as you prod and stretch and tear at the bagel or the loaf. But as bread rests, those starch molecules eventually align themselves into a more rigid form, making the bread firm and tough, even if it hasn’t lost any moisture at all. Minor staling can be reversed by applying a bit of heat and getting those starch molecules fluid again.