You pack brown sugar when measuring it only for the sake of measuring consistency.

White, granulated sugar is essentially dry and flows – not exactly like water, but it flows. If you put it in a measuring cup, it will fill all the space and there will be no voids. If you measure a cup of sugar, you get a cup of sugar.

Brown sugar, with molasses added, is much wetter and tends to clump. If you spoon or dump it in a measuring cup, there will be voids in the sugar, and the amount you measure out today may not be quite the same amount tomorrow or next month. If you press out all the voids by packing the sugar in the measuring cup, you'll essentially get the same amount of sugar every time.

Most of the recipe writers in this country have agreed upon a measure of brown sugar, packed, as the universal standard. They figure if their cup of brown sugar is packed, and your cup of brown sugar is packed, you will have the same amount of brown sugar as them, your recipe will turn out wonderfully, and you'll sing their praises (and buy their cookbooks) forevermore.

A recipe that measures brown sugar by weight (necessitating a kitchen scale) will obviously turn out the same amount of brown sugar each time, whether you pack it or strew it about willy-nilly on the scale.