iPhone blocked by scammers after entering their iCloud

The only way for you to prove anything and have the activation lock removed is to provide Apple with the original receipt from Apple or an authorized retailer.

If you can't do that, there is nothing anyone can do for you.

Do NOT pay the scammers. They're not going to remove the lock on your phone.

If the scammers have put the phone into Lost Mode - as seems likely - Apple might refuse to unlock the phone even if you have proof of original purchase from Apple or an Apple Authorized Reseller.

This may turn out to be a very expensive lesson as to what not to do with your next phone.