Clutch packs inside the differential create additional resistance. This resistance is always present. It is called preload. So, whenever one side wants to start rotating faster than the other, this resistance (preload), in addition to the traction present at the wheel has to be overcome before a tire can spin.
Traction and preload have to be high enough to keep tires from spinning but low enough to still allow tires to rotate at different speeds in a turn. Since the preload has to be kept low enough to allow safe cornering the slowing effect on wheels that want to spin is marginal. It works in easy off-road conditions and on mildly slippery roads. For serious off-road use and very slippery roads (snow, ice) limited slip is not powerful enough.
Remember, it limits slip but it does not prevent it completely. Only a differential lock can do that.