Dr. Edward Harrison gave the "definitive" answer/solution to Olber's Paradox, in his 1987 book "Darkness at Night: A Riddle of the Universe".
While it is an open question wether the Universe is infinite in extent (space), it is finite in time...i.e., it had a beginning (The Big Bang). We can only look back a finite distance (our Cosmological Horizon), so the light from any stars existing beyond the radius of the Hubble Sphere has not had a chance to get to us yet. Combined with the fact that stars themselves have a finite life-time, there is simply not enough visible stars in our observable universe to make the night sky bright.
As Chalmoth pointed out above, the expansion of the Universe also has the effect of red-shifting any distant luminous objects. Even though stars did not exist at the time of Recombination (Surface of Last Scattering), even the unbelievably intense, incandescent light from this epoch (approx. 370,000 years after Big Bang) has been red-shifted to such low frequency/long wavelengths that it is no longer in the visible spectrum...hence the Cosmic Microwave Background Radiation.
Finally combined with observation that the expansion of the Universe is now accelerating, eventually all galaxies (with the possible exception of the local galaxies that are gravitationally bound with the Milky Way) will pass beyond our particle horizon, and will forever become unobservable. Note that I do not subscribe to the so-called "Big Rip". So, in theory, billions of years from now, our Milky Way will truly become an "Island Universe", just as it was once thought of, up until the early twentieth century.