This advice is offered by Atticus Finch to Jean Louise Finch. The idea is metaphoric. Atticus does not want Jean Louise to literally climb inside someone's skin, but rather to experience how someone else is treated. This idea of the skin being the outward representation, the body of the person upon which others react. Atticus is essentially telling Jean Louise to consider how she would be treated if she looked a different way, for example, if she was inside black skin or very dirty skin, she would be treated much differently and she must understand others from that point of view.