It describes the weather patterns of March in the Northern Hemisphere. It means that:
* March begins with fierce, stormy weather, like a lion. Think strong winds, heavy rain, and perhaps even snow.
* March ends with milder, gentler weather, like a lamb. Think sunshine, warmer temperatures, and the first signs of spring.
The saying is a bit of folklore, and while it's not always completely accurate, it does capture the general shift in weather patterns that often happens in March.