How should I proceed after getting in an accident?

I was stopped at a cross walk in a school, and a car hit me from behind. The driver and I exchanged info, and he asked if we could skip the insurance and pay directly. I said that I needed to talk to talk to someone first and I would contact him.

Relevant info: my car is a 2015 Honda Fit that has 5100 miles on it. There was damage to the bumper (small crack and the whole right side is pushed up/out). I have a clean driving record. I live in Massachusetts.

What are the pros/cons to skipping the insurance route? What if I did contact my insurance company? Would it end up hurting me? Would my car end up as having been in an accident (carfax)?

Any advice? Thanks!