Cecil’s answer seems a tad simple and doesn’t account for the red shift in light from far away stars. How about this answer instead from minutephysics http://www.youtube.com/watch?v=gxJ4M7tyLRE ?
original post
Why is the night sky dark? - The Straight Dope
The night sky is dark because if it was light we would call it day. 
We now return you to your regularly scheduled science.
I still think Cecil’s right. I disagree with the video that the red shift trumps the finite age.
Of course, for the deep space pics, we set Hubble to record infrared. But this doesn’t mean that stars at the extreme edge of visibility are red-shifted to invisibility. Are the most distant galaxies red-shifted to invisibility? My guess is “no”. Please disabuse me if wrong!
But regardless, would it matter what the red shift is, if no stars are old enough to be that far away and still be visible?
So, there are basically two reasons why not (why we can’t see them). Either is sufficient; neither is more important, except that (I believe) one kicks in before the other (age), so that would be the more significant one.
Cecil could have said “If the age of the univers wasn’t the limit, then there’s another factor that would eventually kick in: red shift, in an expanding universe.” But did he have to, to answer the question? Once a question is answered, there is probably an infinite number of other reasons. Need we list them all?
While visible light is red-shifted into the infrared, isn’t ultraviolet and other higher-frequency radiation also red-shifted into visibility?
I didn’t find Cecil’s response to be entirely convincing. A candle’s glow isn’t cast very far. Interstellar dust does glow. The two “simple” expectations you might have don’t appear to contradict his conclusion (and in fact the first seems to support it). Perhaps it is because he hesitated to explore the physics too much, and his dismissal makes much more sense once you appreciate the physics of the situation.
Light will travel forever. The reason a candle’s light appears to fade out is because it spreads over a wider and wider area. But that’s because it’s a single point source of light. The interstellar dust hypothesis that Cecil rejects would be more like having a candle at every dark point in the sky.
Again? Sheesh.
You don’t need Big Bang/expansion arguments to explain it.
Take a decent size chunk of the universe, e.g., a billion cubic light years across. Note that the incoming energy is equal to the outgoing energy of this cube. Our local cube is incredibly cold and dim. So it’s putting out a tiny amount of light (in terms of density), hence it is receiving a tiny amount. The far off sky is dark because our local sky is dark. (Ignoring rare nearby anomalies like our Sun and the Milky Way. Most spots in the Universe aren’t lucky enough to be in such bright locations.)
Olbers’ math falls apart when he assumed that the chunks of the Universe 10s, 100s, etc., of billions of light years, are still pumping out energy at the same rate forever. No chunk of the universe can produce such energy for a long, long time. Hence, in a non-Big Bang/no expansion universe, those far off areas are dying embers. In order for Olbers to have been right, each chunk of universe needs an infinite supply of energy and therefore would glow unbelievably (to say the least) bright on its own.
This has been repeatedly published and talked about in Astronomy circles. I have no idea why it’s still considered a “paradox”.
How come?
The whole point of Olber’s paradox was to point out that the dark night sky is evidence that the universe can not be eternal and infinite. Olber was absolutely correct about that.
Remember that at the time (1823), conservation of energy hadn’t been established. No one knew what made the Sun or stars shine. Blaming him for not knowing about concepts that no one had even formulated yet is completely unfair.
To expand on my last post, here’s a timeline of some fundamental physics concepts Olber didn’t have access to:
1823: Olber’s paradox first described by Olber
1840: Olber dies.
1842: Conservation of Energy
1850: First law of thermodynamics
1850: Second law of thermodynamics
1906-12: third law of thermodynamics
1935: Zeroth law of thermodynamics
1900ish: Quantum mechanics
1905: Special theory of relativity
1916: General Theory of relativity
1929: Expansion of universe
That’s a pretty big handicap, compared to what we have today.
This would be true because the universe is homogeneous on large enough scales. One billion-light-year chunk looks the same as any other billion-light-year chunk.
This isn’t sufficient to resolve Olber’s paradox, however. If the universe were static, with stars that didn’t burn out, as time went on, more and more stars would become visible, until the night sky was bright.
It’s true that it’s no longer a “paradox” and it was never a strict logical paradox to begin with. But when Olbers brought it up that we wouldn’t have a dark night sky if the universe was infinite and unchanging it flew in the face of what everyone knew HAD to be true at the time! So it was a paradox in the sense that someone would eventually come along and show why it was possible to have a dark night sky and an eternal, unchanging universe.
Funny how that worked out, eh?
Just to note: “Olbers” not “Olber”. Hence “Olbers’ Paradox” not “Olber’s Paradox.”
Note that Olbers came up with the problem in 1823. People knew enough basic Thermo then to worry about the source of the Sun’s energy. So he would have known that far off stars would have burned out, etc. (Although at a much faster rate than what we know now.)
I think there is some basic confusion and really the question should be broken down into two parts.
The first part of the question is why doesn’t the light from distant stars/galaxies (EBL - extragalactic background light) fill up the night sky such that it is as least as bright as the surface of a star/galaxy? This is the question the one that Olbers’ paradox poses. The answer to this question is the one that Cecil gives, i.e. that the oldest stars have a finite age and that light could’ve only traveled a finite distance to reach us from them. Redshift does reduce the amount of energy received from the EBL, however redshift by itself cannot resolve Olbers’ paradox (there exist cosmological models in general relativity where space is expanding and hence light is redshifted, but where Olbers’ paradox is still in theory an issue) and it cannot be considered the main factor.
What Cecil does get ‘wrong’ is choosing the light travel distance as the distance to the furthest stars: usually their distance is stated in terms of the proper distance which is considerably greater.
The second part of the question is why aren’t we fried to a crisp by the cosmic background radiation (CBR)? This is completely outside the scope of Olbers’ paradox as Olbers could never have envisaged an event such as photon decoupling whereby all points in the Universe emitted a brief intense flash of light. The answer to this entirely is that the wavelength of the light emitted has been redshifted into the microwave region of the em spectrum due to the expansion of the Universe and this is what the video that the OP posted is referencing. To claim that redshifting overall is the dominant factor is highly subjective as without the finite age of the stars the energy flux at any given point is potentially infinite.
Actually I should change the last sentence is the 2nd paragraph to “redshift by itself does not necessarily resolve Olbers’ paradox”