The sheepskin lining on the underside of the saddle skirts has a dual purpose. It provides extra cushion between the saddle and the horse, and it also helps to keep the saddle blanket or pad in place.
Sheepskin first started appearing on western saddles around 1880. The extensive amount of sheep in the American west during that period made it a readily available material.
Sheepskin can either be natural or synthetic. Natural sheepskin, which usually is a golden color, but can also be creamy white, is more expensive and is usually thicker and more durable, if well cared for. But, as a natural fiber, it is susceptible to munching by moths and mice and their ilk. Synthetic or "fake" sheepskin is much less expensive and doesn't attract critters, but it isn't as plush and will wear out faster.
Whether you have natural or synthetic sheepskin on your saddle, it's important to take care of it. Keep it clean and avoid letting the sheepskin side of your saddle touch the ground at all costs. Sheepskin is a magnet for every kind of debris - sticks, bugs, dirt, water, burs - all of which will irritate your horse when the saddle's on his back.
Sheepskin will wear over time with heavy use and may have to be replaced. Replacement is definitely a job for a qualified saddle maker.