This is an expression which owes its origin to a story from Aesop’s Fairy Tales. In the story, a lion and three of his friends – a fox, [a donkey], and a wolf – go hunting. After they make a kill the lion says that he is going to keep three fourths of the meat for himself. The reasons he provides for keeping the “lion’s share” are the following: one fourth as his just share, one fourth for his lioness and cubs, and the other one fourth for his courage. The lion then goes on to inform his friends that he is prepared to part with the remaining one fourth provided one of them challenges him to a fight and defeats him. The three friends refuse to take him on and as a result leave the entire kill to the king of beasts. “Lion’s share” as you can guess from the story means the larger share of something.

Another example:

Since Richard owns 60% of the company, he always gets the lion’s share of any profits.