Why is it called a terry towel?
The name “terry” comes from the French word “tirer” which means to pull out, referring to the pile loops which were pulled out by hand to make absorbent traditional Turkish toweling. - source
Old and used: Some animal shelters take used clean towels. Call first because shelters can get overwhelmed with donations.
For towels in excellent condition, consider giving to a charity that can pass the towels on to their (human) beneficiaries.
There are some organisations that do textile recycling and welcome old towels. Please wash first.
Many vets are happy to receive CLEAN used towels at their practice. Check with your nearest vet and wash them before giving away.
If the suggestions above are not suitable or convenient, throw in trash. Do not put in blue bin.