You can get to the heart of the paradox quite easily, actually. But it never talks about infinite brightness of the night sky -- "just" that every portion of the sky (day or night, looking at the sun or elsewhere) should be as bright as the sun. (This is also enough to kill you outright)

First, suppose you're in an infinite universe. Stars, each roughly as bright as the sun, are scattered throughout it.

But you don't see a distant star as brightly as you do the sun! The star radiates energy uniformly in all directions, so its brightness (the amount of energy it radiates onto us) decreases with the square of its distance. So a star twice as distant is only a quarter as bright as the sun. So are we saved?

Not really. The (expected) number of stars increases with the square of their distance, since we're assuming a uniform distribution. So we expect four times as many stars when we double the distance. At X times the distance from the sun, we expect to find X2 stars, each radiating 1/X2 times as bright as the sun. This exactly cancels out, so we end up expecting the brightness of the sun impinging on us from every area of the sky. (All this can be made precise; trust me).

What "night sky"? (Almost) any point of it should be as bright as the sun!

You may have noticed that this is not quite true. Here are some reasons why not, along with some "reasons" which don't manage to explain why.