LSD is a mechanical version of traction control, limiting wheel spin on an axle in two and four-wheel drives. We can notice the main advantage of a LSD when it is compared to a typical standard differential. Imagine a scenario whereby in the case of the standard differential one wheel has no contact with the ground. In this scenario, the contacting wheel will remain stationary, and the non-contacting wheel will rotate at nearly twice its intended velocity! Therefore the torque transmitted will be zero and the vehicle will remain stationary.
This sort of situation is rather rare in everyday driving, but in more extreme situations such as in motor sports or in off-road driving this situation is highly undesirable and does occur. Using an LSD limits the velocity difference between the two wheels and allows for torque to be transmitted correctly given that some friction is available on at least one of the driving wheels.