Why do some think that the Greek city-states might have asked Persia to intervene in their feuds?

Answer 1

A couple of hundred in Asia Minor were in the Persian Empire. Each of the Greek city-states, wherever they were located - in Europe, North Africa or Asia, followed its own interests, banding together in leagues as it suited them in their inter-city disputes, and changing leagues as it suited their interests.

Those cities in Asia Minor were of course restive under Persian rule, but many had cut deals in their own interest. The Persian preferred tactic was to bribe city ruling classes to go along with the Persian idea of peace and prosperity under their its rule, only resorting to force when this failed. This was witnessed by large Greek city defections at battles during the Ionian Revolt.

And the Revolt itself arose when Greek tyrant of Miletus, Aristagoras, got into trouble with the Persian governor over a failed attempt to pillage the Greek city-state of Naxos, and to save his own skin, set about stirring up the Ionian Revolt.

Answer 2

The Greek aristocracy were easily bribed, and so it was usual for some to switch sides.

For example at Lade, half the Greek side stood back and did not engage, leaving the rest targets for easy defeat.

For a couple of years before the Xerxes invasion of mainland Greece, his emissaries were circulating through the cities with bags of gold. At Plataea there were over 40,000 Greeks from the northern cities fighting in the Persian army - a third of its strength.

So there were two factors - amongst the Greek city-states there was usually warfare between them on a small or large scale, and the Persians gave the opportunity to both pursue these enmities and make personal profit.