What happens if an underage person is driving a car without a licence and they crash into another car?

You need to report this to your insurance company in any case in order to keep coverage for future events.
The other driver seems to be engaging in extortionate behavior if what you say is true. He may have committed a crime. Even thought the police probably will not take jurisdiction over the crash itself because it seems to have happened on private property.

Your daughter may be facing some sort of DMV consequences if she violated a rule. It is very unlikely she has committed a crime.

They could file a report, regardless of whether or not you pay them. Nothing could stop them.

This matter should be turned over to your automobile liability insurance carrier. Advise them that your daughter had your permission to drive the vehicle, assuming this is the truth. Your insurance company has an obligation to deal with this matter and defend and indemnify your daughter. You should not have to deal with the adverse party directly.

Legal Disclaimer:

Mr. Lundeen is licensed to practice law in Florida and Vermont. The response herein is not legal advice and does not create an attorney/client relationship. The response is in the form of legal education and is intended to provide general information about the matter within the question. Oftentimes the question does not include significant and important facts and timelines that, if known, could significantly change the reply and make it unsuitable. Mr. Lundeen strongly advises the questioner to confer with an attorney in your state in order to insure proper advice is received.

If your daughter has coverage you must report the accident to your carrier. If not, you could settle out of pocket with the other driver, but have him get several estimates.
Don't let his threaten you with a police report. The rights of the parties are the same whether or not a report is filed.