The phrase "to stay with" means to continue doing something or being in a particular state, condition, or situation. It can also mean to remain loyal, faithful, or committed to someone or something. For example, if you are asked to stay with your children while they take a bath, it would mean that you need to be present and supervise them during their bath time. Similarly, if you are working on a project, "to stay with" can imply continuing to work on the same task or job until completion.