The soul is a metaphysical concept and so requires a metaphysical explanation.
The soul is not part of the body. It exists outside of this reality in a realm with far less form and definition than this one. It is essentially the connecting thread between us and God.
Humans engage in something we call spiritual growth. This is an extension to learning and explains why certain of us are far more capable than others. When we learn, we store the memories of the thing we learned from in our brains, for ready access. The soul, among other things, is the repository of past learnings from past lives.
They can't be stored in the brain, because it dies at the end of our lives. It can't be anywhere physical because physical information is limited by the rules of physics, it wouldn't be able to follow us to the next life, assuming the next life is even in this realm.