In this case "Dresser" is wardrobe person. When Smith was 17, he got a job as a dresser for the Stratford Shakespeare festival in Stratford, Connecticut, and he got to meet some big stars like Katherine Hepburn. But what this memoir is really about is how the beauty of Shakespeare's plays and poetry helped him through a troubled childhood. His younger sister was born crippled with cerebral palsy and when the mother was overwhelmed (which was often) it was his job to take care of his sister and change her messes. He loved his sister and anguished over her. It was a sad situation.