It is a normal closing for good wishes. In the middle of winter, 'stay warm' would be taken as a wish that you keep warm now that the cold weather is upon us. He could have said "good health" or "be happy" - not so much commanding you to do so, but expressing his best wishes to you that you stay in good health and/or be happy. I would consider it more personable and friendly than something more business-like, such as 'Best regards'