To take the lion’s share is to take the larger part or even the majority of what is to be apportioned out.
There are many tales from ancient times that tell of a group of animals going hunting and the lion using his strength and position as king to claim all the spoils.
The story is included in Aesop’s Fables and here is one version.
The Lion went once a-hunting along with the Fox, the Jackal, and the Wolf. They hunted and they hunted till at last they surprised a Stag, and soon took its life. Then came the question how the spoil should be divided. “Quarter me this Stag,” roared the Lion; so the other animals skinned it and cut it into four parts. Then the Lion took his stand in front of the carcass and pronounced judgment: “The first quarter is for me in my capacity as King of Beasts; the second is mine as arbiter; another share comes to me for my part in the chase; and as for the fourth quarter, well, as for that, I should like to see which of you will dare to lay a paw upon it.”
“Humph,” grumbled the Fox as he walked away with his tail between his legs; but he spoke in a low growl. “You may share the labours of the great, but you will not share the spoil.”
This expression has stood the test of time and centuries later is well understood and common in the media.
From The Times in London: “Les Misérables …… accounted for the lion’s share of cinema attendances.”
From India’s Daily Bhaskar: “Chief minister’s hometown Jodhpur got lion’s share in the state Budget announced by him on Wednesday.”
From the LA Times: “At Freddie Mac and Fannie Mae, which control the lion’s share of the government-backed loans …”