Don't make it sound bigger than it was. The 'island' is one of a pair of uninhabited islets (rocks, really) with a total area of 4 hectares.
There is some confusion over their status because in the past, they weren't considered important. They're between the Turkish mainland & a group of islands which were ceded by Turkey to Italy in 1912, & then by Italy to Greece in 1948. Their status under the original cession was vague. Turkish officials recognised them as Italian between WW1 & WW2, but that recognition does not have the force of a treaty.
The incident was sparked by a Turkish merchant ship running aground, & refusing Greek assistance. A weak Turkish prime minister tried to make political capital out of the not quite certain status of the islets*, & the intervention of a group of Greeks who hoisted a Greek flag, followed by some Turkish TV journalists who landed & filmed themselves pulling down the Greek flag & raising their own, broadcast live, escalated it. There was no 'invasion': some marines were landed by both sides & glared at each other across the channel between the islets.
The USA spoke to both sides & after discussion via US diplomats both agreed (quite readily, apparently) to withdraw their troops & ships & return to the status quo ante.
*My feeling is that it's not definite, but the Greeks probably have the better claim.