On the surface, George take cares of Lennie because he promised Lennie's Aunt Clara he would do so, but there are deeper reasons for his actions. Lennie helps George as much as George helps Lennie.
As George says early on in the story, he and Lennie have what most migrant workers don't, which is companionship. They have each other for friendship and to lean on in adversity, which makes facing their hard lives easier.
Although George often gets angry at Lennie and complains at having to take care of him, in reality, the two men have a strong, caring bond, and George loves Lennie dearly as a friend. Each complements the other: George is the type of person who likes to be in charge, and because Lennie is mentally handicapped, there is no quarrel about George making all the decisions for them. Lennie, in turn, needs George's guidance if he is to survive at all. Additionally, while George is small and wiry, Lennie is big and strong, offering George a measure of protection.
Lennie gives George's life meaning and purpose. Lennie is a friend he can confide in and a person who he knows needs him. Lennie also makes it possible for George to dream about a better future in which they can buy a small farm together. Such a dream becomes bitter without someone to share it with. When George kills Lennie out of mercy, he loses everything important to him.