This section is from the "The New Student's Reference Work Volume 5: How And Why Stories" by Elinor Atkinson.
One-half of the answer to this is: As bears sleep most of the time in a warm place in winter, they do not need as much food as when they are active. The other half is that they eat themselves. In the summer months they eat greedily, a great deal more than they need at the time. This extra food is stored up in their bodies in thick layers of fat. This fat keeps the sleeping bear warm. Food is fuel. As he is very warm in his blanket of fat, a bear needs less food. And, gradually, he eats the blanket. The fat is absorbed into the blood to feed all the other tissues of the body. In the spring the bear comes out thin and poor.
Many other animals hibernate in the winter. Snakes store up fat to live on. You see there is little food for them in winter, so Nature taught them how to stock up their internal pantry shelves for hard times. When you are sick and can not eat as much as usual, you, too, live partly on the fat stored in the body. That is why you become very thin. And that is why, when you begin to get well, you are as "hungry as a bear."