On the surface, Lennie crushes Curley's hand because George tells him to fight back. So, he does this out of fear, obedience to George, and in self-defense. Lennie, himself, has no real purpose in crushing Curley's hand other than for those reasons. So, within the world of the story itself, this was simply a case of two men getting into a fight and one of them getting injured. To protect Lennie, Slim convinces Curley to say he got his hand caught in a machine. It is reasonable to assume that Curley agrees to this to keep the peace with Slim and the other men but also because he doesn't want to look like he was bested by another (weak-minded) man.
Although there is no personal purpose (from Lennie) in crushing Curley's hand, the purpose or significance of this act does have deeper implications in terms of the sociological aspects of life on the ranch and psychological aspects of Curley himself.
When Curley's hand is crushed, he essentially becomes handicapped and this puts him in a similar position to Lennie who is mentally handicapped. The worth of a ranch "hand" is gauged/valued on his ability to work. With a crushed hand, Curley's value in terms of labor and economy diminishes. This says something about the harshness of ranch life and the devaluation of human life when it is gauged primarily on labor value. But also, individually, the loss of the use of his hand (even if temporary) must add to Curley's insecurity and low self-esteem. The clear connection is with Candy who, being old and having lost a hand, fears the day when he will no longer be useful. The crushing of the hand increases Curley's anxiety and anger towards Lennie which makes him all the more determined to get Lennie when he kills his wife.
There is a lot of symbolism dealing with hands in this story. George is described as having "sharp, strong features" with "small, strong hands." Lennie is "a huge man, shapeless of face" who drags "his feet a little, the way a bear drags his paws." When they first meet Curley, "his hands closed into fists." Each description of hands reflects some aspect of the character's personality. George is small and sharp. Lennie is like a bear; his hands are potential weapons. Curley is confrontational.
The swamper tells George that Curley keeps a glove filled with vaseline on his left hand. "Curley says he's keepin' that hand soft for his wife." As the hand is crushed, not only does Curley become less valuable as a worker (a social implication); he also becomes, at least symbolically and psychologically, sexually impotent because he claimed that he used the hand to sexually please his wife. The hand is a symbol of strength, value, and masculinity. When it is crushed, these values are diminished within the social context of being a ranch "hand" and within the psychological reaction of feeling economically, and perhaps sexually, impotent.