As stated in the item's description, this item can be used to rename a Hunter'sPet. It is consumed when used.
When it is used, a "Rename" option will appear in your currently active pet's right-click pop-up menu. If you swap pets before you use the rename option, it is lost and you must wait out the cooldown before renaming it again.
Note that this item is not required to name your pet from its Pet family name(e.g. 'Bear', 'Cat'). When you first tame your pet, it will already have the "Rename" option.