"To damage the car" means to harm or cause harm to the car, typically resulting in physical or mechanical problems that make the car less functional or attractive. Full definition
The card comes with an auto rental insurance coverage, in case you are driving a rented car while traveling, and accidentally damage the car in some way. (financeproductreviews.com)
Property Damage Liability — provides financial protection for damages your car causes to someone else's property. (accuquote.com)
Although not required, it is still better to be prepared since no one can predict the extent of damage a car accident can bring. (autoquotenow.com)