The technical name for this is interproximal reduction (IPR). It is a very common procedure done with Invisalign. The purpose of this stripping is to make some space for crowded teeth to straighten out. Think of this as flossing with a sandpaper-like metal strip. The orthodontist will move the metal strip or use a very thin disk on the hand piece in between your teeth to take away about 0.1-0.4 mm of enamel. This is such a small amount that it is not noticeable and you will not experience any increased risk for cavities in the future.
Interproximal reduction is most often needed when you have crowding of your teeth. To straighten teeth out, the orthodontist needs some space to help align the teeth, and this space is gained in between the teeth. The video above shows the stripping done with a thin disk.
The slenderizing is done quickly and you will experience no pain in your teeth. It's a weird feeling but definitely not painful. You do not need to be numb at all. However, you might have some slight irritation around your gums at the appointment. Your gums might feel sensitive for 1 day after the slenderizing. If you are wearing Invisalign trays right after the slenderizing, you might experience more discomfort, since the trays can hit the irritated gums. Again, this will go away in about 1 day but please discuss your concerns with your orthodontist.