Epsom salts are high in magnesium, a key component to healthy muscle movement. Magnesium is one of the “electrolytes” that is advertized in sports drinks and such, and it is an essential mineral (metal) for movement of muscle fibres.
Sore and taxed muscles (say, after a hard soccer game or a bicycle race) will be soothed by having magnesium come to them directly through the skin when you soak. This is much more effective and efficient for the muscles than having magnesium absorb through the digestive system. It is also great after a deep tissue massage (meant to break up knotted muscle tissue) to soak in epsom salt, as the electrolytes help the muscle tissue heal without knotting.