How do I repair this damage to my driveway?
The technical term for what happened to the concrete on your driveway is that it "spalled", and sealing it with caulking isn't going to do much toward repairing it at this point.  What you'll want to do is clean out all of the loose debris (a shop-vac and wire brush work pretty well) and then pressure wash it.  After it is completely dry, inject or force epoxy down into any crack that extends down from the damaged area. Then apply a concrete bonding agent, fill it with a patching concrete and trowel it smooth.

Note - depending on the condition of the rest of the driveway, you might want to look into applying a concrete sealer to extend its life.