For the most part, this is not the place to come to experience Thai food or culture. For a slightly more authentic experience (and cheaper than the well-decorated cafe/restaurants by Haad Rin beach), patronise the more modest street carts where you might see some Thais eating.
The best area for cheap eats is definitely the food carts in Thong Sala, the main town on the island. In the evening you can eat at the night market (from 40 baht/meal (usually curried meat with rice or fried rice or pad Thai), soups from 30 baht, pancakes from 20-30 baht, meat on sticks 10-30 baht/stick.) Free Wi-Fi in a roofed hall. Across the road from 7-Eleven in Chalok Lam are also some Thai street stalls where you will find locals eating. In front of Tesco you can buy sticks with fried meatballs for 5 baht/stick and at the junction next to entrance to Tesco parking lot you can buy in the morning (07:00-09:00) sticky rice with pork/chicken/livers packed in banana leaves for 10 baht for a small portion. There are also a few street stalls. There is also a small market next to 7-Eleven in Thong Sala in the direction of Haad Rin (after Walking St traffic lights) with Thai meals for 30-50 baht.