* Atmosphere and Scattering: Earth's sky is blue due to our atmosphere. Sunlight is scattered by the molecules in the air, and blue light is scattered more than other colors. This phenomenon is called Rayleigh scattering. Without an atmosphere, there's nothing to scatter the sunlight, so the sky appears black.
* Lack of Light Reflection: An atmosphere also acts as a reflector, bouncing light around. Without an atmosphere, the light from the sun directly reaches the lunar surface, but there's nothing to reflect it back to make the sky appear bright.
Why you see stars during the day on the Moon:
Because there is no atmosphere to scatter light, the stars are visible even during the lunar "day." You can see the stars directly without the interference of scattered sunlight.
