Essentially my question is what happens after you get your wisdom teeth removed?

Not all are the same: Wisdom teeth can be simple extractions or more invasive procedure if the teeth are impacted. Impaction means they are buried either covered by gum or both gum and bone. The more invasive the more swelling and discomfort afterwards. The doctor should prescribe pain medication and advise you to ice your face. From my experience the ice makes a very big difference.

Healing: After your surgery you will notice a hole where once the roots were and some swelling for a few days. Sometimes if you lose the blood clot you can develop a dry socket which is painful and needs treated by your surgeon. Nothing needs done once you are healed.