Answer:
This is an idiom that is no longer considered good medicine. It comes from a culture from long ago before we understood pathogens, disease, and our immune systems.
When you are ill with a cold, other upper respiratory infection, and/or fever, eat sensibly when you are hungry and stop when you are satisfied. Drink plenty of liquids. Rest in bed. Treat the symptoms with over-the-counter medicines. Ask your pharmacist for product recommendations for your specific symptoms.
Don't rely on this old saying as medical advice.