Claiming allows a contact record to change ownership without the need for an Admin to change it for a user. A contact record can be claimed one time only. Once claimed, the claim button will no longer be available and an Admin will have to change the ownership of the record.
The Claim button can be found on top of the contact record, along the other action buttons.
Claiming the record will change the ownership to the current user. Once claimed, the button will disappear.