Why isn't the universe lit up? Sun the sun and many other stars shine their light outwardly, and light travels very long distances, why is space dark? Why isn't it all lit up due to the stars? 
 A: There are two effects here. The first one, in our neighborhood and galaxy is answered by the fact that light sources open up uniformly  and the intensity falls as 1/r^2 

The closer the source , the more light hits our atmosphere and reflects back into our eye. At night if there is a moon, there is some intensity because the moon is close. In a moonless sky the intensity reaching us from the stars is very small and thus the darkness at night.
On top of that it is the effect that on the way from the stars to the earth there is very little matter, atoms, molecules dust , so space cannot be lit up even by the sun, as the atmosphere is lighted. 
Then we come to the second effect which was discussed in the comment to your question, Olber's paradox .
If the universe is flat and infinite, it means that there would be an infinite amount of light arriving at earth from and infinite number of heavenly bodies and the source dispersion becomes irrelevant, the night sky should be all lit up, and even the few particles per cm^3 would recieve an infinite amount of radiation and passing it along light up "empty space".

The darkness of the night sky is one of the pieces of evidence for a dynamic universe, such as the Big Bang model. If the universe is static, homogeneous at a large scale, and populated by an infinite number of stars, any sight line from Earth must end at the (very bright) surface of a star, so the night sky should be completely bright. This contradicts the observed darkness of the night.

