This isn't a question with an "answer" as such, so it is a good opportunity to indulge in a bit of informed speculation. Most historians, when discussing the question, tend to put it in economic terms for very good reason. The art, literature, philosophy, rhetoric and even sciences of classical Athens were fueled by money. The Parthenon is a stunning monument of human ingenuity, but it was also one made possible by great wealth. This wealth is often attributed to Athenian imperialism, which is undoubtedly true but also somewhat puts the cart before the horse. Imperialism was highly rewarding to Athens, but was only possible because it had reached such a position in the lead up to the Persian Wars, which gave it the prestige and organization neccesary to bring the Aegean into a sphere of control. Broadly speaking, I would say there were two factors that allowed Athens to reach this position: the first is its geographic location, which put it in favorable place regarding Aegean trade routes, And, unlike Corinth, the mountain ranges surrounding it made it relatively cut off. The political organization was due in many respects to the Peisistratids, who continued and expanded the set of institutional reforms started by whatever person or group of people are responsible for the reforms attributed to Solon. Most important, the sum total of the Solonic and Peisistratid reforms was that Athen ahda sort of unfed "national" identity much stronger than in other parts of Greece. This allowed reformers like Cleisthenes to expand the citizen base considerably, and incidentally to also expand the base of those eligible for military service.

That is a very, very brief explanation of why I think Athens was in the position it was that allowed it to become such a major power after the Persian Wars. A great source to look into this is Pomperoy et al Ancient Greece: A Political, Social, and Cultural History, which is something of an introductory textbook but is quite good in most respects.

Now why did Athens become a major cultural force? Part of this is that its wealth acted as a magnet for talented Greek individuals who wanted to make a name for themselves--think of it as equivalent to New York or London. But I think an interesting angle is actually the Persian actions in the eastern Aegean. If you look at early Greek thought, a disproportionately large number of major thinkers come from eastern Asian Minor (modern Turkey): Thales, Heraclitus, Pythagoras, Anaximander, etc. It is possible that they didn't feel fully comfortable in eastern Anatolia and so left for Greece, and the natural destination was Athens. It is worth noting that this basic effect of non-Athenians moving to Athens was very common--Gorgias, Herodotus, and Aristotle, for example, were not Athenian. It is possible that Persian incursions and rule pushed the center of gravity of Greek culture from eastern Asian Minor to mainland Greece, where Athens was the most natural fit.

This is not an answer, but more of a comment on some logical issues with the question: it seems beset by some complex (and problematic) assumptions:

a) the history, artwork, etc. we currently have of the 400 BCE period is representative of the "creativity" of the city-state during that age

b) the history, artwork, etc. (or lack thereof?) from other time periods are equally representative of the "creativity" (or lack thereof) of those periods

c) we can measure creativity reliably and validly from type and quantity of historical artifacts and records we have of Athens at multiple points throughout history.

An actual historian can probably make an awesome statement about how much historical "coverage" we actually have, continuously, for ancient Greece, but it's not going to be anything near complete, and I suspect it's not even going to be enough for us to infer that we really know how much "creativity" was going on during all the different periods to which your question has implicitly compared 400 BCE.

And then there's the issue that creativity is something on which there's a lot of disagreement--we don't have consensus on its boundaries or functioning, or even what it really is (though that's not to say we know nothing about it...). The concept itself is probably strongly culturally/socially defined.

Going out on a limb, I suspect the original question finally boils down to something like, "How does one explain the apparent creativity, defined by [insert criteria], that the historical record and remaining artifacts seem to show, which seems to be relatively greater, due to more [insert evidence] for the Athens 400 BCE period than for other Athens periods?"

I'm by no means an expert but Tom Holland gives some reasons in his book "Persian Fire" firstly he states that one of the ways the oligarchich families would attempt to one up each other was to patronise the arts, as Athens was growing rapidly in status at this time there were a lot more resources and financial incentives available to the artists and architects and more and more Greeks were paying attention to their works (historically Athens had been a small backwater town focused on agriculture). When Athens won its democracy it resulted in a sweep of social conciousness and pride which in turn led to more plays, monuments and buildings. Athens was already becoming the forefront Greek city thanks to its trailblazing and newly booming economy. Couple this with their leading role in the Persian Wars and you have an environment that inspires creativity set against a backdrop that gives it great cultural significance, ensuring it was preserved.

Greece was just coming out of the dark ages after the Dorians invaded and all record keeping essentially stopped. The Dorians were savage and illiterate and imposed that onto the people. This lasted for several hundred years. After this ended and things regained a bit of normality, there was an influx of wealth, and this wealth funded the artists and scientists to do what they did.