Why is the sky dark at night?
I'm going to respond to (v1) of the question, which asks why the night sky is dark (black and unlit) compared to the day sky even though there are many light sources at night. The updated question references Olber's paradox, which has been answered many times before.
Like most things we see in everyday life, there are a number of reasons contributing to this. I'll go through each of the leading effects and what happens if we get rid of them.
First is my number one reason that the sky is dark at night simply because it doesn't require us to change the fundamental nature of the universe in order to imagine a universe without it. It is because most of the light from other stars is blocked by dust and gas clouds between us and the other stars.
Here's a picture:
(source: wikimedia.org)
This is an image of the Milky Way as we see it from Earth. Without dust and gas, this would be so bright that the night sky would be lit up as well. This is mostly due to how bright the central bulge would be, but I'd be lying if I said the rest of the disk isn't bright as well. So most of the light that comes from other stars is blocked by dust and gas. Blocked of course meaning reflected, refracted away from us, or absorbed and retransmitted as invisible radiation.
Second, cosmological expansion. This gets the number two spot because of how important it is. The expansion of the universe causes light to be shifted to lower frequencies. The farther away it is, the more it is shifted. You have to change the nature of the universe to do it, but let's consider what it would be like without expansion: There's this thing called the Cosmic Microwave Background (CMB), which is background radiation that comes in almost equally from every direction. When it was produced, this radiation would have been as bright as if everywhere around you was the surface of a star (a reddish-orange star). Without expansion to redshift this to the microwave range, that's how bright the night sky would be now. That's significant right? Apparently not significant enough. Because of expansion, there is a cosmological event horizon, a distance at which things recede from us faster than light. Because of this, we can't see anything beyond that distance. So without expansion, all the stars and galaxies and light sources within our particle horizon but outside our event horizon would be visible (and not redshifted); so the night sky would be even brighter.
Third is the fact that the universe isn't infinite in age (technically we found that out because of the expansion of space, but this is my answer and I'll make it it's own thing if I want to). It has a finite age and that means light can only propagate a certain distance within that time. Therefore, we have a particle horizon, which means that there can only be a finite number of stars visible to us. This is great news because otherwise there would be an infinite number of stars visible to us and they would come in from every direction, which would make the sky as bright as the surface of the sun. That would be bad.
Those are the main three. Dust, expansion, and a finite age of the universe. You could say that expansion should be the frontrunner, but without expansion we really wouldn't be here and the universe would be fundamentally different. It's at least possible that our galaxy could have formed and given us a line-of-sight view to the galactic core, so dust and gas wins for actually keeping light from hitting us that could have in the first place.
Who is interested can find detailed information at wiki, or here
The problem is known (as you added in your edit) as Olbers' paradox, and was posed already in the mid 1500's, by Johannes Kepler in 1610 and even later by Edmond Halley in the eighteen century, and curiously, even the novelist an poet Edgar Allen Poe anticipated possible explanations as to why the night sky is dark But it was popularized as a paradox by German astronomer Olbers in the nineteenth century.
It says that if the universe is infinite and static, then at any given angle from the Earth the line of sight will end at the surface of a star. An infinitely old universe means that there has been plenty of time for the light from every star that has ever shined to reach our eyes, there should be stars everywhere, and the sky at night should be just as bright as when the Sun is up.
The explanation for why the sky is dark instead of comes from recent discoveries about our universe made since Olber's time. From what was known up to about the nineteenth century, it seemed seemed very reasonable that the universe was infinitely old and unchanging, and in such a universe, Olber's paradox is a real problem.
We now know however, that the universe is not infinitely old and static, the universe had a beginning: the Big Bang. This has important implications for Olbers Paradox. Because the universe has a finite age, one reason our night sky is dark is that many photons have not had time to reach us
But the BB presents another paradox: it states that the early universe was flooded with photons. At this time the cosmos was truly bright. Given these hot, bright early conditions, shouldn't there be a luminous curtain of light behind every star and galaxy we see?
The fact is, this curtain of light is there, but our eyes cannot see it. Due to the expansion of the universe, the wavelengths of these hot, early photons have been stretched over 1,100 times longer than their original wavelengths, the high-energy backdrop of the early universe, is filled with cool, microwave photons (the CMB), invisible to the human eye.
But, after all Olber was right as was Poe: they thought that the night sky should be as bright the noonday Sun, and in fact it is. What they didn't realize was that the night sky is only dark to us. If they had eyes sensitive to microwaves, there would never have been a paradox. One might argue that also frequencies above the visible spectrum are redshifted and become visible, but one should consider the net balance. And then, of course, we must consider the relevance of the excellent reminder of Kyle to Jim, that most part of radiation is reflected away from us, scattered by dust.
update
EDIT: I've seen both answer may be acceptable, but what is the main contributing phenomenon?
your friend thinks:
...the fact that in a finite amount of time ..light can travel only a certain, finite, distance, [determines the effect that] we cannot see, if they actually exist, stars farther than such distance. (and he implicitly thinkss) that, within this radius, there aren't enough stars to fill the night sky, and that's why we have darkness without sun's light.
If you have correctly reported your friend' thesis, then your friend is definitely wrong, because within that radius there are enough stars and we can't see their light because of dust
Your thesis is partially wrong, as it is a minor element of the issue, but that is not the main cause, but I think we can safely conclude that: you win the bet by default
The following passage is extracted from Stephen Hawking's book "A Brief History of Time":
In fact, various contemporaries of Newton had raised the problem, and the Olbers' article was not even the first to contain plausible arguments against it. It was, however, the first to be widely noted. The difficulty is that in an infinite static universe nearly every line of sight would end on the surface of a star. Thus one would expect that the whole sky would be as bright as the sun, even at night.
Olbers’ counter-argument was that the light from distant stars would be dimmed by absorption by intervening matter. However, if that happened the intervening matter would eventually heat up until it glowed as brightly as the stars. The only way of avoiding the conclusion that the whole of the night sky should be as bright as the surface of the sun would be to assume that the stars had not been shining forever but had turned on at some finite time in the past. In that case the absorbing matter might not have heated up yet or the light from distant stars might not yet have reached us. And that brings us to the question of what could have caused the stars to have turned on in the first place.
The passage gives a brief account of the correct statements in Jim's and bobie's answers.