The verb translated as “stay awake” is the same term Jesus used in Matthew 24:43. Taking seriously the likelihood of a home invasion implies that you take the necessary precautions. In a world without deadbolts and alarm systems, that implied staying awake (or taking turns staying awake). It equally implied that you were ready for instant action, without the loss of time implied in having to get dressed. The picture is one of being alert.
15 (“Behold, I am coming like a thief! Blessed is the one who stays awake, keeping his garments on, that he may not go about naked and be seen exposed!”)