A pure white piece of what looks like skin just peeled off in the inside corner of my mouth where my teeth are, why?

Trauma? did you bite your cheek or lip? Irritating, hot, spicy food? Used a new toothpaste or mouthwash? A white colored tissue is not unusual with these things.
Irritation. The thin mucosal lining of the inside of the lips and cheeks is very susceptible to chaffing, chemical change such as new mouthwash or toothpaste and to extreme heat such as hot fluids or food. If it persists have your dentist look at it to rule out other causes.