Submitted by Jim Early, Orange County, and answered by Dr. Roger Blandford, Richard Chace Tolman Professor of Theoretical Astrophysics and Executive Officer for Astronomy; and David Hogg, Caltech graduate student in physics.
A similar question was asked by Shawn McCord, age 8, of Covina.
This is one of the oldest and most fundamental observations in cosmology, known as "Olbers' paradox." After all, if the universe is infinite and filled with an infinite number of stars, shouldn't every line of sight from Earth hit the surface of a star somewhere? Everywhere you look, you ought to be looking at something as bright as the surface of the Sun.
The sky is dark because the universe is of finite age, born roughly twelve billion years ago in the Big Bang. Because light travels at a finite speed, the part of the universe we can observe is not infinite. In fact, the radius of the visible universe is given by the distance light can travel in twelve billion years: twelve billion light years. Not every line of sight hits the surface of a star; in fact most get to the edge of the visible universe without encountering anything at all. So the night sky is dark, to human eyes.
On the other hand, although the night sky looks dark to us, it is actually very bright with microwaves which make up the cosmic background radiation, the relict light from the Big Bang. For the first three hundred thousand years after the Big Bang, the universe was so hot and dense that it was opaque and glowed like a star. Because light travels at a finite speed, objects observed at a very great distance are also being seen as they appeared a long time ago. So the rays of light which come from the edge of the visible universe were emitted when the universe transitioned from its opaque to its transparent state. If the universe were not expanding, this "surface" in time would glow bright like the surface of a star.
However, the universe is expanding, so the light waves are stretched to longer and longer wavelengths as they travel, and are now stretched into microwaves. To an astronomer with a radio telescope that is capable of detecting microwaves, the night sky does indeed appear as bright as the surface of a star—not because the visible universe is infinite and filled with stars, but because, early on, the universe itself shone bright like an immense star.