Kepler saw this as an argument for a finite universe, or at least for a finite number of stars, but the argument is not convincing as will be shown below.
One explanation attempt is that the universe is not transparent, and the light from distant stars is blocked by intermediate dark stars or absorbed by dust or gas, so that only light from a finite distance away can reach the observer. However, this reasoning does not resolve the paradox. According to the first law of thermodynamics, energy must be conserved, so the intermediate matter would heat up and soon reradiate the energy (possibly at different wavelengths). This would again result in uniform radiation from all directions, which is not observed.
Another resolution that has been offered points to the fact that every star contains only a finite amount of matter and therefore shines only for a finite period of time, after which it runs out of fuel. This theory seems to have been first suggested by poet and writer Edgar Allan Poe. However, the paradox stands if one assumes that stars are constantly being created randomly across the infinite universe, shine for a finite period, and die.
The paradox is resolvable in a variety of ways. If the universe has existed for only a finite amount of time, as the prevalent Big Bang theory holds, then only the light of finitely many stars has had a chance to reach us yet, and the paradox breaks down. Alternatively, if the universe is expanding and distant stars are receding from us (also a claim of the Big Bang theory), then their light is redshifted which diminishes their brightness, again resolving the paradox. Either effect alone would resolve the paradox, but according to the Big Bang theory, both are working together; the finiteness of time is the more important effect.
In addition to the expansion causing the frequency of the light to fall below observable frequencies, the fact that more distant objects are receding faster from each other than are near ones, means that after a certain distance, the expansion of the Universe occurs faster than light travels through it, meaning that the light emitted in one place simply will never make it to the distant regions of the cosmos. For example, if we were to send a radio signal to the most distantly observed galaxies, the signal would never arrive there, because the rate at which the distant galaxy and ours are receding from each other excedes the rate at which the radio waves travel through space. Example 2: At one point in the early Universe, two galaxies at opposite ends of the cosmos were 18 million light years away from each other, meaning that light emitted from one should take 18 million years to travel between them. However, due to the expansion of space, the distance that the light had to travel to get from one galaxy to the other has continuously increased, which increases the transit time of the light between them, such that we see the light they emitted then, several billion years later. As the expansion continues, the ability of light to cross the void is lost, another reason for the darkness of the sky.
In fact, the darkness of the night sky is nowadays taken to be evidence in support of the Big Bang theory.
Another explanation, which does not rely on the Big Bang theory, was offered by Benoit Mandelbrot. It holds that the stars in the universe may not be uniformly distributed, but rather fractally like a Cantor dust, thus accounting for large dark areas. It is currently not known whether this is true or not, although recent satellite studies have found the Cosmic microwave background radiation is isotropic to 1 part in 10000.
Note: The name Olbers already has an "s" on the end, so the possessive is Olbers' (or, alternatively, Olbers's). It is incorrect to write Olber's because his name was not "Olber".