When you first begin whipping cream, millions of tiny air bubbles get whisked into the liquid. The cream becomes frothy and lightens. If you stopped whisking now, the bubbles would eventually work their way out and the cream would become a uniform liquid again.
But if you keep whipping, something more happens. Bubbles are still getting whisked into the liquid, but now the action of the whisk also starts stripping away the protective outer membranes on the fat globules. This allows the fat to join together and gradually form protective bubbles around the tiny pockets of air.
Technically this is an emulsion - air suspended in liquid and held stable by fat - and is also why you need cream with a relatively high fat content (at least 30%) in order to whip cream. The less fat, the more those globules are stretched thin and the harder it becomes to make a stable emulsion.
All this being true, whipping cream still feels a bit like magic to us!
(We consulted Milk by Anne Mendelson and On Food and Cooking by Harold McGee for this information.)