Stars do glow during the day, but we can't see them because of the glare of sunlight. When the sun is up, the blue color in sunlight gets scattered all over the atmosphere, turning the sky the familiar bright blue color. This blue light is much brighter than the faint light coming from the stars, so it prevents us from seeing them.
If you were standing on the Moon, for instance, where there is no atmosphere, you would see the stars both day and night.