If you accidentally damage someone else's property when driving you are financially responsible for compensating them for the damage. If you want to pay out of pocket rather than use your insurance they would ideally accept this offer. However, they should take the car to a few body shops because the damage might cost more than you expect.
From their end it really doesn't make any difference whether you pay for the damage or your insurance company does at the end of the day. Hopefully, you can both come to an agreement on a good body shop to take the car to, the car will be repaired to their satisfaction, and they'll be on their way.
If you can't come to an agreement you'll need to file a claim. You won't have any deductible to pay as the damage goes under your Liability coverage where there isn't one. Whether your insurance goes up on renewal depends on your insurance company. Some don't count little accidents less than some threshold like $600 against you. Many car insurance companies also have accident forgiveness where they don't charge more for having one accident on your record.