I guess we'll have to ask the person who's business card it's on. Most pet sitters are bonded, especially if you'll go in to someone house. It's a nice thing to assure the pet owner. BUT, I was not aware of any licensing of pet sitters, unless it's through another outside group....
They might be referring to a business license, which you don't need for a pet sitting business. You might want to join a petsitters group like Petsitters International (www.petsit.com). They usually have a lot of good info for getting your business set up and to make sure that you've covered all of your bases.
They are probably referring to the licence you need from Louisville Metro Animal Control. You need a license for boarding (babysitting is considered the same thing as they are in your care during that time) You do need this as the fines are steep for non compliance.
As a petsitter, I don't petsit in my home. I go to their's for scheduled visits. So Lucy, if you are taking pets into your home, then yes, you would need some type of license or permit. You may even need to look into zoning issues with your home. All of these issues is why I don't bring people's pets into my home - too many liability issues.