The Color of Water is James McBride's autobiographical story of growing up in an extraordinarily non-traditional household. The entire book recounts his journey from confusion and disquiet about who he is to a relative peace with his circumstances. Young James hasmany things in his life which make him different from others, and many of them are puzzling to him because his mother, Ruth, reveals next to nothing about her past. It is only as Ruth tells her story that the questions in James's life get answered and he is able to, as your question says, "come to terms with who he is as a person."
James is one of twelve children and grows up in a chaotic household run primarily by his rather eccentric and disorganized mother. Though she was relentless about her children's educations, the rest of their lives were in disarray. His father was black and his mother, Ruth, was white, though she lived her life trying to make the world see her as a light-skinned black woman--and all of this in the midst of significant racial tensions and the rise of "Black Power." Ruth was born a Jew in Poland but ended her life as a Christian woman. She never spoke of her Jewishness to her children. All of these elements combine to create James's crisis of identity.
Ruth's virtual silence about her own identity created dramatic questions and doubts in James about his own identity. So much of his life was just not quite right, and he eventually realizes that he must learn his mother's story in order to answer the questions of his own life, race, and identity.
He is right. As Ruth tells him bits and pieces of her life and as he discovers other facts on his own, James comes to reconcile the confusing elements of his life. In the end James knows he is the biracial son of a Christian father (and stepfather) and has a converted Jew for a mother.
The specific answer to your question is that he comes to terms with his identity only as he learns about his mother's identity; it is only then that the disconcerting elements of his life make sense. That is why half the story is written in Ruth's voice and words while the other half is written from James's point of view.