Malcolm X presumably took up his work for the same reason anyone decides to advocate on behalf of a group of people- because he saw (and experienced) suffering in the world and wanted to change that. Though we often refer to Malcolm X as a Civil Rights activist, he was firmly against the Civil Rights Movement because it focused on integration of the races. It is important to bear in mind that Malcolm was shaped by a long legacy of the oppression of Black Americans. During his time with the Nation of Islam, he spoke publicly on the state of race relations and believed that white people were the source of evil in the world.
Earl Little, Malcolm's father, was a preacher and member of the Universal Negro Improvement Association. Before his birth and all throughout his childhood, his family were the target of racial violence. In 1929, their home was set on fire by white supremacists, and though Earl Little's death was officially ruled a suicide, it is suspected that he was murdered for the color of his skin and his beliefs.
The surname Malcolm chose for himself was a testament to his struggle to overcome the legacy of Black oppression- "X" represented the lost name of his African ancestors. He is an icon of a turning point in American society and for having been the public face and voice of the Nation of Islam until his break with them in 1964. He believed that civil rights was not just about the equal treatment of white Americans and Americans of Color, but also about recognizing the legacy of American slavery as criminal.