Packing brown sugar just means pressing it firmly into the measuring cup. We find it easiest to fill the measuring cup first, then press the sugar gently down using the back of a spoon. You don't need to use a lot of force. The sugar will compact and leave a quarter-inch or so gap at the top. Use the spoon to fill this in, press the sugar down again, and then repeat as needed until the measuring cup is filled.
What we're doing is pressing out the pockets of air that get trapped between the sticky grains of sugar. If we didn't do this, our volume measurement would be off. We only want to measure the sugar, not the air. Packing the brown sugar is also a way of making sure the measurement is consistent from batch to batch and from cook to cook.
(Image: Emma Christensen)