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Lennie kills Curley's wife by shaking her so hard that her neck breaks -- remember, he is a big strong man. He does not kill her on purpose. He is only shaking her like that because he wants her to stop yelling.
She is yelling because he is holding on to her hair and won't let go. She asked him to stroke her hair but then he grabbed in a bit. When she started to complain, he panicked. He was afraid that George would not let him be part of the dream anymore if Curley's wife got mad at him.
So he's shaking her to try to protect his part in the dream. But he kills her and the dream too.