Licensed, means some state agency has given you a license. Even a driver's license. So you have at least, a driver's license. Bonded, means someone has sold you a bond. There are HUNDREDS of different bond types, and what exactly it means, depends on the type of BOND you bought. Bonds are NOT transferrable from one client to another, so for most practical purposes, if there's no bond in your hands, the person is not bonded for YOU.
Insured, means you bought some kind of insurance. For a pet sitter, maybe it means CAR insurance, so they can drive to your house. Maybe it means LIABILITY insurance, maybe it means workers comp, in case the pet sitter employee gets bitten by your dog.
You will have to ASK the pet sitting business, EXACTLY which license, what kind of bond, and what kind of insurance they carry.