Perhaps it's a good time to use the technical terms of Adipose Tissue instead of Fat cells as well as Lipids instead of Fats.

If you look at the White Adipose Tissue (WAT) on the right, you will see that it's largely a storage bag with a single lipid-filled space. On the other hand, the Brown Adipose Tissue (BAT) has many small lipid droplets, many mitochondria (the energy converters of the cell), and many capillaries. Hence its brown colour.

Interestingly, Brown Adipose Tissue is more closely related to Muscle Tissue than it is to White Adipose Tissue. So, calling it all fat, gives an entirely wrong impression.

Put simply, WAT stores lipids while BAT is thermogenic tissue. When activated, BAT cells take lipids and run them through the mitochondria to generate heat rather than synthesise ATP (the body's chemical energy storage molecule).

So that's how infants can generate heat without shivering and thus, avoid hypothermia.