There are many possible explanations which have been considered. Here are a few:
The premise of the second explanation may technically be correct. The philosopher Immanuel Kant suggested we live in an Island Universe in that most of the Universe is empty and black but there is a grouping of stars, where we are and which we called the Milky Way Galaxy. However, in the 1920 Edwin Hubble had shown that there were many galaxies outside of our own Milky Way. Now we know that there are at least as many external galaxies as there are stars in the Milky Way (roughly 100 billion). We also know in principle that there are many more galaxies which we cannot see. Thus the number of stars, finite as it might be, may still large enough to light up the entire sky, i.e., the total amount of luminous matter in the Universe is too large to allow this escape. The number of stars is close enough to infinite for the purpose of lighting up the sky.
The third explanation might be partially correct. We just don't know. If the stars are distributed fractally, then there could be large patches of empty space, and the sky could appear dark except in small areas.
But the final two possibilities are are surely each correct and partly responsible. There are numerical arguments that suggest that the effect of the finite age of the Universe is the larger effect. We live inside a spherical shell of "Observable Universe" which has radius equal to the lifetime of the Universe times the speed of light. Objects more than about15 billions years old are too far away for their light to have reached us yet.
Historically, after Hubble discovered that the Universe was expanding, but before the Big Bang was firmly established by the discovery of the cosmic background radiation, Olbers' paradox was presented as proof of special relativity. You needed the red-shift (an SR effect) to get rid of the starlight from the visible. We now actually view the effect of expansion as stretching of the wavelength of light as space stretches. Thus visible light is shifted to longer and longer wavelengths, thus the "red shift". This effect certainly contributes. But the finite age of the Universe is the most important effect.
References: Ap. J. _367_, 399 (1991). The author, Paul Wesson, is said to be on a personal crusade to end the confusion surrounding Olbers' paradox.
_Darkness at Night: A Riddle of the Universe_, Edward Harrison, Harvard University Press, 1987