Douglass focused on others at first for several reasons. One reason is that he was still a child and although he was subject to the same discrimination as other slaves, he was not subjected to the extreme brutality and discrimination of adult slaves. By recounting the things he witnessed as a child, Douglass shows how his thoughts and philosophies were influenced.
I was so terrified and horror-stricken at the sight, that I hid myself in a closet, and dared not venture out till long after the bloody transaction was over. I expected it would be my turn next. It was all new to me. I had never seen any thing like it before.
(Douglass, Narrative of the Life of Frederick Douglass, gutenberg.org)
Another reason is that Douglass, as a child, simply didn't have as many personal experiences to explain. Instead, he uses the experiences of others as examples of discrimination, showing how racism in those days was endemic and unthinking. No one thought differently about slaves, and so no one thought to protect them or make their lives better.
Another reason is to show how slave children had little or no family structure in their lives. To the owners, all slaves were interchangeable, and so children and parents were sold, killed, and replaced without concern for the children's mental state. Because he never had a family life, his father and mother vanishing from his life at an early age, Douglass felt no connection with them as parents, only as ideas, and this sort of social detachment is shown as "normal" for slaves; the slave society developed under explicit status as possessions instead of people, and Douglass shows that through the experiences of himself and of others.