Why Space is Black

Space is look black on the surface of the moon in both day and night, because moon has not, its own atmosphere, therefore there is no scattering of light.

The daytime sky is blue because when light from the sun, enters the earth atmospheres it gets scatter. (In earth atmosphere there are various molecules, dust, moisture etc., present which creates sunlight scatter in all directions).

In the night time space looks black from earth because there is no light source available on the atmosphere of the earth to scatter.

Why it is not brightening by the light of other stars.

Due to Expansion of our universe, star go farther away from us in a continuous manner, so the source of light is also moving away from us, due to which the wave length of light became longer. This means the light from the stars that are moving away from us is shifted towards red, and at some points its wavelength become so longer that it is no visible at all.

Illustration diagram is given below,