The Goodwill location where I live collects worn out and stained fabric items for textile recycling. I label a kitchen trash bag 'textile recycling' then when it is full I tie it off and drop it off like any other donation.
You can even include old shoes! But anything that is ruined with a hazardous substances like motor oil, gasoline, etc. should not be included.
Goodwill sells this scrap fabric to the companies who process it to make new clothing, blankets, industrial rags, filling for furniture, insulation, etc.
Adding an additional statement just in case: I know a few people who are anti-Goodwill because they saw an image posted online about horrible groups to avoid. These graphics contain few, if any, factual statements. Legal information about charitable groups is freely available to the public, so it's always a good idea to independently verify.