What could be causing my toothpaste to strip the gums from inside my mouth?

Gum infection. Should see your denist, sounds like gum infection that usually needs treatment.
An additive. Different toothpastes have different additives. If you gums are irritated change toothpastes. Preferably a non tartar toothpaste.
Triclosan. We no longer recommend any toothpastes that contain triclosan. Triclosan is an antibacterial agent used in soaps (hand soaps and shampoo). Somewhere someone decided that it would be a good idea to add it to toothpaste. It is a bad idea because the mouth is too sensitive, as you discovered. We recommend sticking to regular fluoridated toothpaste without whiteners, tartar control, or triclocsan.