Why would an ankle sprain cause lower leg pain?

Sprain is injury of. The ligament, it could strech or rupture completetly which also injury to the nerve supply the are of the sprain that lead to painful situation.
Possibly add'l cause. Some ankle sprains may in fact not be an ankle sprain but a specific type of fracture that results in the fracture of the fibula high up the lower portion of the lower leg (just below the outside of the knee) and this would require evaluation with an additional radiographic series. There could also be other tendinous injuries.