This bit of folk wisdom has been bouncing around for centuries. This advice may have evolved from the idea that illnesses could be classified as either low temperature (those that give chills, such as a cold) or high temperature (those with fever). With chills, it sounds reasonable to feed a person's internal fireplace with food. The logic follows that when an illness raises the body's temperature, cutting back on the "fuel" should help. However, scientific evidence doesn't endorse this advice -- many illnesses must simply run their course.
Nevertheless, if you're stuck in bed with a cold and a loved one brings over your favorite healthful foods, it's still OK to chow down. Alternatively, you may lose your appetite while fighting a fever-based sickness. When you're sick, it's okay to miss a meal or two as long as you are keeping up with fluid intake.