Because the widow is alone and Young Fu is her only son, he is allowed to complete his apprenticeship while living in a small rented room with her, rather than living in the shop, a plot device which allows us to see more of the city than might otherwise be the case.
The widow's solitude and Young Fu's status as her sole child allow him to live with her, rather than in the workshop, which provides a plot device that enables us to see more of the city.