Why did Persia at the time think it can invade Greece and make it part of the persian empire?
Greek lands back in the day were geographically more extended than today. And this applies to Greek mainland itself, excluding colonies. Namely, Asia Minor across the Aegean was a component of Greece, seeing how much of it had been hellenized. And the Iranians successfully conquered and integrated that eastern half of Greece into their empire. So it is essentially the western half of it which for the most part remained independent.