This is the second Military Policy, along with Raid, designed to improve your raiding. That is, gaining some one-time yields from pillaging in enemy lands, without trying to conquer them. The advantage of pillaging Districts, as opposed to improvements, is that a single district tile may be pillaged multiple times, until all its buildings are destroyed. And this could be advantageous if your army isn't strong enough to really invade and face the city defenses, or if you aren't actually after the city itself.
A good strategy is to amass an army of light cavalry units with the Depredation Promotion, slot both Sack and Raid, then go raid the enemy for a significant boost to your yields! This particular strategy is good for protracted wars, where your goal is not that much taking territory than hindering your enemy.
Remember that if you actually plan to annex other cities, pillaging their districts may be more trouble than it's worth, simply because repairing them back to functionality after you conquer the city may take huge work and render the conquered city much less productive.
Since most soldiers weren’t (and aren’t) well-paid, looting was the primary source of income … for those lucky enough to survive the battles, plagues, and punishments. Unlike pillaging the countryside, the chance to sack a built-up place offered much more than mere sustenance. During the Middle Ages, a lord’s encouragement to sack a town, city, or district was the main way to insure loyalty and further willingness to fight. Even the gentry joined in the fun, often claiming the most priceless pieces and best folk to hold for ransom. The bigger and richer the city, the more to go around; thus, some places like Rome were sack-prone (410, 455, 546, 1084, and 1527 AD).