There are at least two answers to the question of why Curley started a fight with Lennie. Curley has a big inferiority complex because of his small size. Steinbeck does not explain this but has Candy explain it to George, since most of the exposition in this novella is in the form of dialogue, which will make it easy for Steinbeck to convert his book to a stage play. Curley takes an immediate dislike to Lennie because Lennie is so much bigger than he is. He attacks Lennie to show how tough he is and what a good boxer.
The other answer has to do with Steinbeck's reason for inserting this fight at all. Steinbeck wanted to establish that Lennie had extraordinary strength, partly to explain how easily he could kill Curley's wife by accident. However, the author could not show Lennie working in the fields because he deliberately avoided having any outdoor scenes in which the men and teams of horses would be shown harvesting grain. He confined his story mainly to the bunkhouse and the barn. This was because he intended to convert his book into a stage play almost immediately, and there would be no way of showing Lennie tossing four-hundred-pound bales of barley onto wagons, as George claims he can do when they are being interviewed by the boss.
The fight between Lennie and Curley is intentionally restricted to a small area because two actors could not be moving all over a stage cluttered with bunk beds and crowded with men watching the fight. Therefore, Steinbeck has Lennie grab Curley's hand and crush it while both men remain in approximately the same spot.
The author's purpose in inserting this fight was mainly to show how strong Lennie really was. It is one thing to say that a character has tremendous strength, but it is far more effective to demonstrate it. After the injured Curley has been taken off to Soledad for medical attention, Slim voices Steinbeck's implicit message.
In a moment Slim came back into the bunkhouse. He looked at Lennie, still crouched fearfully against the wall. "Le's see your hands," he asked.
Lennie stuck out his hands.
"Christ awmighty, I hate to have you mad at me," Slim said.
After it is discovered that Lennie has killed Curley's wife in the barn, George realizes that Lennie has become a menace he can no longer control. He realizes that Lennie's interest in soft little animals such as mice, rabbits, and puppies has evolved into a sexual attraction to human females. George remembers the incident in Weed with a new understanding of Lennie's motivation.
"I should of knew," George said hopelessly. "I guess maybe way back in my head I did."
George is the only person who knows where Lennie will be hiding. It is at this point that he makes the decision to kill his friend as an act of mercy. He canot help him escape the angry mob as he helped him after the incident in Weed. The Weed incident was relatively trivial, but if George helped Lennie escape this time he would be an accessory to an apparent murder.
The fight with Curley, which left him with a mangled hand, will also serve to explain Curley's intense hatred of Lennie and his desire to have him killed in the most slow and painful way. This in turn will enhance George's motivation to kill his friend mercifully in order to save him from a horrific lynching.