The easiest way to answer that is to watch some horse racing. A leg up is receiving assistance from a groom or other trained person to get into the saddle.
It's called a leg up, because the assistant places his or her hands under your leg and lifts upwards as you hold the withers and pull. Both elements take a certain amount of practice. (The way jockeys are thrown into the saddle is amazing, but works primarily because jockeys tend to be very small and light).
Leg ups are commonly used if riding bareback. Side saddle riders also often get a leg up because this method of mounting puts no weight on the left stirrup (side saddles are more prone to slipping when mounting from the stirrup).