Does a person have to file a claim with insurance if the person who hit the car offers to pay for damages?

You do not absolutely have to file a claim with an insurance company if you and the other driver can resolve the matter without insurance and you can get your car repaired.  It is always a good idea to have the other driver's insurance information.  Many times drivers say they are going to pay for repairs without using their insurance and then they change their minds and disappear or stop taking your calls.  Good luck to you.