This question seems to confuse the order of cause and effect. The battle of Thermopylai established the image of the Spartans as selfless, death-defying badasses. The men who set out to the pass did not have this reputation. They were heroic warriors because they fought at Thermopylai, not the other way around.

Unfortunately, no source tells us about the decision-making process from the Spartan perspective. All we have is Herodotos' account of the Greek reaction to the news that Xerxes' army was advancing into Greece. However, Herodotos gives us a lot of relevant information, so that we can contextualise the battle that made the Spartans famous.

When the Persian Wars began, the Spartans were the most powerful people in Greece. There is nothing in Herodotos' account to suggest that this was due to their superior fighting skill. Much more than that, they seemed to rely on a large citizen militia, a system of unequal alliances, great PR thanks to their tyrant-deposing campaigns abroad, and occasionally some good old-fashioned trickery and deceit. As leaders of the Peloponnesian League, they no longer had any rivals on the Peloponnese, and could command a substantial number of other Greek city-states in war. Many Greeks therefore regarded them as de facto "leaders of Hellas" (Hdt. 5.49.1). They did not need to showcase their military power; it was already well-established.

When those of the Greeks who were willing to resist the Persians gathered to discuss strategy, the Spartans took it for granted that they would get supreme command. Herodotos praises the Athenians for accepting this, despite their own rising status in the Greek world; the Athenian decision showed their commitment to the greater good. The Argives and the tyrant Gelon of Syracuse were not so gracious. They refused to submit to Spartan leadership, saying they would only help the Greeks if they could be in charge of the army and fleet. The Spartans, however, would not back down, and so the support of two of the most significant Greek states was lost.

This episode shows that the Spartans could not simply assume that everyone would follow wherever they led. They had to prove that they were worthy to be in charge. To make things worse for them, the first action of the anti-Persian alliance under their command was a complete fiasco. At the request of the Thessalians, the Greeks decided to guard the pass at Tempe (between Mount Olympos and Mount Ossa) with 10,000 hoplites, but when they discovered that the pass could be turned, they immediately withdrew the army again. The Thessalians, feeling betrayed, went over to the Persian side. This was a serious loss for the Greek alliance - and a blow to the prestige of the Spartans. Their leadership was seen to have fallen short; they hadn't even sent a king to Tempe, putting a mere polemarch named Euainetos in charge of the allied army. Many suspected that the Spartans would be all too happy to sacrifice all Hellas north of the Isthmus of Corinth, retreating to their own power base on the Peloponnese and fighting from there. A more convincing display of Spartan commitment to the Greek cause was needed.

This is the context in which we need to see the decision to march to Thermopylai. Admittedly, Herodotos does not say it was a Spartan decision; "the Greeks" decided to fight there, as it was the most defensible spot. It had previous been fortified by the Phokians to keep the Thessalians out, and none of the southern Greeks knew about the path that could be used to turn the pass until they got there (Hdt. 7.175.2). "The Greeks" therefore resolved to send "their army" to Thermopylai. There is no explicit mention of the Spartans in the council scene in which this plan is forged (Hdt. 7.175-177). Yet they had the command, and it must be assumed that they had a big hand in the decision.

The trouble was that Xerxes had quite rudely invaded right when the Greeks were celebrating the Olympic Games, and the Peloponnesians were celebrating the sacred festival of the Karneia. It was generally regarded as a bad time to march out. The force of about 7,000 hoplites that was initially sent to Thermopylai was therefore no more than a holding force. Few of its contributing allies sent more than a few hundred men. The Greeks did not think Xerxes would move quickly; they expected to have time to send their full levies to the pass in time for the decisive showdown. Thermopylai was in any case the perfect place to make a small force count for a lot, so that there was little concern that the army would come to harm.

Even so, the Spartans sent 300 picked men under the command of king Leonidas, since they knew that half measures would not do to convince their allies of Sparta's good intentions:

The Spartans sent the men with Leonidas on ahead so that the rest of the allies would see them and march, instead of switching to the Persian side like the others if they learned that the Spartans were delaying. At present the Karneia was in their way, but once they had completed the festival, they intended to leave a garrison at Sparta and march out in full force with all speed.

-- Hdt. 7.206.1

This is how Leonidas and the 300 Spartans (possibly along with 700 perioikoi, and certainly along with an unknown number of helots) ended up at the pass. It's important to stress again that they did not expect a battle. When they discovered that Xerxes was actually right on the doorstep, the little army's generals were torn:

The Hellenes at Thermopylai, when the Persians drew near the pass, fearfully took counsel whether to depart. The rest of the Peloponnesians were for returning to the Peloponnese and guarding the isthmus, but the Phokians and Lokrians [whose land was right behind the pass] were greatly angered by this counsel. Leonidas voted to remain where they were and send messengers to the cities bidding them to send help, since they were too few to ward off the army of the Medes.

-- Hdt. 7.207

There are a couple of important points in this passage:

• The Greeks were clearly not prepared to fight with the tiny advance force they had at the pass.

• It was the Phokians and Lokrians, not the Spartans, who insisted they should stay and fight - entirely out of self-interest, since their lands would be ransacked if the Persians were allowed to advance (as indeed happened after the Persian victory).

• While Leonidas agreed that they should defend the pass, he also insisted that they should send for reinforcements immediately. There is no hint that he had the intention, let alone hope, to fight to the death for glory's sake.

On the morning of the third day of the battle, the Greeks learned that the pass had been turned. This was the point where the Spartans made the fateful decision to fight to the death. However, Herodotos admits that he did not exactly know how or why this happened (Hdt. 7.219-221). Either the army simply split between those who decided to stay and those who wanted to leave, or Leonidas himself ordered the allies to leave; if the latter, he either did so because he thought it dishonourable to leave his post, or because he put faith in an oracle that said Sparta could be saved if one of its kings sacrificed himself, or because he believed Sparta would get more glory if they stayed while ordering the others to leave. Herodotos puts some weight on the last option, which suggests a consciousness of a chance to earn "bragging rights" - but only because they had already marched in defence of the rest of the Greeks, and because their decision not only meant the sacrifice of their own lives, but also the preservation of those of their allies.

Since this is pretty much all the information we have, modern scholars can do no more than speculate about the assumptions and intentions of the Greeks who stayed to die at Thermopylai. We will never know exactly why they did it. However, one thing is certain: intentional or not, the last stand of the Spartans at Thermopylai must count as one of the most bafflingly effective propaganda stunts in human history. Since Leonidas gave the order, and since the Spartans carefully curated the commemoration of the battle at home and at the site of the battle, they managed to take complete possession of the story. The other Greeks who fought to the death were pushed to the margins of history. No one thinks of the Thespiaians as noble, heroic, superhuman warriors. But this is all in the aftermath. Spartan power in the century after Thermopylai rested to a great extent on the reputation they earned that day, and they milked it for all it was worth - which is why, to this day, we may still get the impression that the indomitable Spartans marched to the pass for no other reason than to be able to brag about what they would do.

Tl;dr: the Spartans fought at Thermopylai to prove their commitment to the defence of Hellas; they succeeded so spectacularly at this that our perception of Spartan history is still warped by it even now.