There are several reasons why a cat makes a purring sound when it breathes:
1. Feeling comfortable and relaxed: Cats purr when they feel comfortable, content, and happy. This usually happens when the cat is being affectionate with its owner, sleeping next to its owner, or eating. This sound is caused by vibrations in the throat and is normal, generally requiring no additional treatment.
2. Self-soothing: When faced with insecurity or tension, cats will also purr to help regulate their emotions and achieve a self-soothing effect.

3. Pathological causes: Purring can also be caused by pathological reasons such as respiratory infections, pharyngitis, or laryngitis. In these cases, purring is usually accompanied by symptoms such as coughing, sneezing, and runny nose. It is necessary to take your cat to a veterinary hospital for further examination and treatment as soon as possible.