The paradox itself is that if the universe is both infinite and eternal then we shouldn't have a dark sky. This is 'trivially' solved by demonstrating that either of those conditions aren't true. The article opts to use the explanation given by Edgar Allan Poe to demonstrate this, which is that the universe has a finite age and the speed of light is finite so there's only a finite amount of observable universe, which gives us a universe which was darker in the past and one that will only get brighter in the future as more of the universe becomes observable. This has some problems of course and the model Poe would have been working with would have been one of a cyclic universe of eternal growth and decay. This leads us to the Big Bang theory.

>> The redshift hypothesised in the Big Bang model [...]

>Doesn't sound to me like it's more than a hypothesis, but I could be wrong.

The way the Big Bang theory resolves the paradox is similar to that of how Poe resolved it, with a finite cap on the age of the universe there's only a finite amount of observable universe, and similar to Poe's explanation it presents a problem in that a younger universe would have been immensely bright. However this new issue is resolved through the explanation of the expansion of space which can be observed through the redshift of distant galaxies.

As we do observe a dark sky we know the hypothesis that led to the paradox can't be true, namely that the universe is both infinite and eternal, so the question is less about why we have a dark sky and more about what possible alternate hypotheses resolve the paradox. While the Big Bang theory is just a theory it's important to remember that proof is reserved for maths, a theory is a hypothesis backed up by observational data. General relativity led to the hypothesis of an expanding universe and this was something that was later observed from redshift measurements and from it we derive the Hubble–Lemaître law, that galaxies are moving away from earth with speeds proportional to their distance, in some cases faster than the speed of light, this alone fully resolves the paradox and crucially the Big Bang theory is not incompatible with this observation.