Did 300 Spartans really stop the Persian empire?

No. The 300 Spartans were part of a force of roughly 7,000 Greeks, they lost the battle, and the Persians went on to conquer Phocis, Boeotia, Attica and Euboea. What turned the war was a naval trap about a month later. A Greek force of approximately 7,000 men, including an estimated 300 Spartans, marched north under King Leonidas in 480 BC to block the pass of Thermopylae. For seven days, three of them in direct battle, the outnumbered Greeks held off the Persian army before the rear-guard was annihilated.

The pass was lost to betrayal rather than to a breach. A local man named Ephialtes, motivated by the desire for a reward, told Xerxes about a mountain path around the pass. His name came to mean "nightmare" in Greek and the archetypal traitor. Leonidas then dismissed most of the army and stayed with 300 Spartans and 700 Thespians, and others reportedly stayed too, including up to 900 helots and 400 Thebans.

The turn came at sea, at Salamis. Themistocles, the Athenian general, sent his servant Sicinnus to Xerxes with a message claiming he was on the king's side, in what appears to have been a spectacularly successful piece of disinformation. The Persian fleet rowed into the narrow straits, where its great numbers became an active hindrance, and the outnumbered Greeks won. The Persians made no further attempts to conquer the Greek mainland.

Even the size of the Persian army is contested: Herodotus claimed 2.6 million fighting men, while modern scholars estimate 120,000 to 300,000.