If you look on the Airbnb Help website it talks you through the steps you need to take to claim on the Airbnb Guarantee @Justine336
Basically as Airbnb advised you contact guest to submit a claim under the guarantee with photo/video evidence you took of the damage before you tried to clean it and let them know you couldn't get the stains out so are hiring a professional carpet cleaner and will be invoicing them for the cost
if the guest doesn't reply or denies Airbnb will adjudicate .