Hi cherin. In this famous speech Kennedy had just made a joking remark about how hot it was and when the applause died down he followed it up with the phrase you quote.. and got another laugh. He is saying that he is the one standing up in the heat making this major speech (i.e. "doing all the work") but he is nearly finished so the audience only needs to stay settled (i.e. "stay cool") for a short time. Like all good public speakers he knew how to inject a casual phrase to connect with a specific audience, in this case mostly college students.
Student or Learner