There was a lot of strategy at play in the evacuation and subsequent destruction of Athens. If we took Herodotus' words at pure face value, the simple explanation would be that it was prophesied by the Oracle at Delphi and Themistocles was clever enough to figure it out. When the Athenians consulted the Delphic Oracle in 480, she told them, among other things, to "trust the wooden walls." As Herodotus tells it, while everyone else thought that meant the defesnive fortifications of the city, Themistocles successfully argued that she had meant the Athenian navy.

As always with oracles in Greek literature, it is sufficiently vague to be interpreted in many ways (basically any wall they thought of would have been made of wood at the time), and modern historians have developed a variety of reasonable explanations for the Athenian strategy.

The first points to note are geography, numbers, and experience. Over the course of 480, Xerxes and the Persians conquered, sacked, and burned their way through any opposition from Thermopylae to Attica. It didn't set a strong precedent for single cities holding their own against the Persian army. Of course, Athens was larger, wealthier, and more powerful, so maybe they stood a better chance. There's still the issue that the whole Greek alliance was generally outnumbered by the Persian invasion force. An empire stretching from Greece to India could just plain muster more soldiers than the Greeks could ever hope for. Of course the full force of the Persian empire was never deployed all in one place, but the fact remains that the Greek Allies were outnumbered by a significant margin.

Then, we face the further issue of geography and internal Greek politics. If you look at a map of Greece, you can see that Athens is located in the southeastern promontory of Attica where the mainland splits from the Peloponnes at the Isthmus of Corinth. The rest of Greece north of Attica was under Persian occupation. The bulk of the remaining allies were all from the Peloponnese, centered on Sparta. The Peloponnesians wanted to mount a defense at the Isthmus of Corinth. It was narrow, defensible, and they could muster a larger force there than they could have at Thermopylae. That strategy (which they ultimately used) left Athens undefended. The Greeks just didn't have the numbers to safely defend Corinth and Athens. They'd have to split their force and the Persians could have chosen to attack Corinth instead. Too many of the remaining allies faced Persian conquest to leave that option open to the Persians, so Athens was left in the lurch.

So the Athenians, significantly outnumbered and on their own were left with a choice. Stockpile supplies, fill the Acropolis with as many soldiers as possible, and brace for a protracted siege, or flee. Either way, their only option to save the general populace was to order an evacuation because, as you said, they couldn't hold the whole city up in the citadel. So there was going to be a mass evacuation anyway and they had no choice but to cede the surrounding city to the Persians while they besiege the Acropolis because defending the whole circuit of the city from such a large force was never going to be practical. If they wanted the people to survive unconquered then the Athenians were leaving either way. At least that's how it is portrayed in Herodotus.

Personally, it doesn't seem to me like their odds were very good if they decided to hold their ground in the first place and that alone would be a valid reason to retreat. However, there was still more strategy at play. The Athenians (in an effort spearheaded by Themistocles himself) had been investing in their naval strength since the first Persian Invasion a decade earlier. The Athenian navy was crucial to any allied strategy to hold land for long because they needed it to act a blockade or attack force against the Persians. The earlier Battle of Artemesium, where the Allied fleet blocked the Persian navy while the Battle of Thermopylae stalled the Persian army proved that it was an effective strategy, but also that the Athenians were a necessary component of that strategy.

Then there's the issue of Themistocles offer to switch sides. Themistocles sent a messenger to Xerxes from Salamis, stating that the allied commanders were arguing among themselves and that he wanted to side with the Persians. The first part was true, the other Allied naval commanders wanted to go help with the defense of Corinth, while Themistocles was advocating to stay at Salamis, where the close quarters gave them the advantage despite the larger Persian fleet. The message to Xerxes served two purposes, one was to lead the Persian ships into an ambush. Thinking the Athenians planned to desert their allies, they would have overwhelmed the other Greek ships, and thus the Athenian offensive came as a surprise.

The second was to lay the groundwork for favorable treatment for Athens if everything went wrong. If the fleet was defeated at Salamis, Themistocles could have argued that the Athenians had already capitulated and tried for better terms than the other Greek cities. At least that's the theory proposed by some historians. Obviously, we can't actually know because Herodotus doesn't explicitly state that motivation. If that really was Themistocles' strategy, then it would only have worked if the Athenians had not put up significant resistance, which thus far they had not.

So, the Acropolis was well suited to traditional Greek warfare and the size of Greek armies, but the Persian Invasion force was much larger than the Athenian defense could ever hope to be, and they had established a pattern of defeating lone cities. Athens was on its own because the other allies had to prioritize the Poloponnese, and their leader was aware that their Navy was not only powerful, but crucial. To sustain that component of the Alliance, they had to flee. Then, to cap everything off, there is a good possibility that Themistocles was putting bets on both parties involved and sought to use the weak Athenian defense to his advantage if everything went south.

Major Secondary Sources:

• The Greek Wars by George Cawkwell
• From Cyrus to Alexander by Pierre Briant
• Encylopaedia Iranica