They wanted to prevent a (hypothetical) monarchist faction from inviting in a member of a foreign royal house and having him elected as President as a step toward making him king.

There was significant fear of an executive turning his office into an autocracy, as had happened in the Roman Republic, had effectively happened in the Netherlands, and would later happen in the French Third Republic. To guard against this, virtually every state had so strictly limited its governor's power that future constitutional conventions would have to enlarge it. Since the Presidency was less limited, supporters of the Constitution had to assuage these fears.

And, while people were definitely aware a natural-born President could become an autocrat, importing a monarch from abroad was relatively common at the time for new realms - see the Congress of Vienna, or Britain's own recent naming in 1707 of Sophia Electoress of Hanover as heir to the crown (though she had a hereditary claim as well). Even in the 1800 Presidential campaign, Adams' opponents spread a rumor that he'd tried to marry his son to a daughter of George III to establish an American dynasty. It wasn't the primary fear in the framers' minds, but it was definitely present.

Article Two of the Constitution states a president must be a "natural born citizen," but doesn't define it further. This has led to some significant debate over its definition and, some contend, does not require a president to have been born in the U.S.