"Grinds" is a local term for food, as in, "Want to get some grinds?" I have heard it used as a verb as well, "Go ahead and grind." (encouraging one to eat) In this piece, "Local" is a euphemism for "inexpensive" or the opposite of "fancy".
Nonetheless, these are places I really like for authentic food and good vibes - recommended no less than the fancy places I've listed, just don't expect white table cloths and shiny silverware. For most, don't expect table service at all. You will get a unique Kauai experience and tasty food.
I'm omitting a couple of "standards" because I find them off-putting; you know those places that are institutions so while the food is indeed good, the staff doesn't try to be friendly, a little conceited by the fame of the place?
CLOSED: Hanalei Pizza (Hanalei) - good homemade pizza from the crust up, good combos, vaguely hippie-style ambiance, and they usually have ready-to-eat slices. The crust is tasty, containing a bunch of things like hemp meal and chia seeds (all written on the blackboard) and I learned from the chef that yesterday's unused dough goes into the mix recycled into tomorrows dough.
Tahiti Nui (Hanalei) - this bar restaurant has been around a long time and has live music several nights a week. I like to go from the music, sit at the bar, and have a Caesar Salad which is very good and a good sized portion. I haven't explored the menu much beyond that but it looks like the kitchen knows what they are doing. Also untried by myself but promising is their Wednesday night luau dinner.
Mermaids Cafe (Kapaa) - a little hole-in-the-wall next door to Java Kai consisting of just a kitchen, you eat on tables out front or on the counter along the side. Menu is mostly wraps: I like the Cilantro Ahi and the Ahi Nori Wrap. A little pricey considering how un-fancy the place looks, but its a good portion well prepared,
Pono Market (Kapaa) - not a restaurant but a fish market counter, I mention them for excellent poke in many varieties and an assortment of other local foods to go.
Tip Top (Lihue) - a new find I've only been there once for breakfast: the macadamia pancakes were excellent. The restaurant is in the middle of what looks like very inexpensive motel in downtown. It's a big place, and was full of mostly locals on a weekend morning. They have a sushi bar which I hear (from a Japanese person) is best on the island (which may not be saying much, haven't tried it). Recommending just on faith that it's worth further visits.
Koloa Fish Market (Koloa) - tiny place just beyond (going makai) the Koloa Old Town shops, it's hard to find -- look for it exactly opposite the Koloa Post Office. Fresh fish, poke, lunch plates, and some desserts. I like the ahi (or other fish, as available) wasabi (there are a few options for preparation) lunch plate: two good sized pieces of fish cooked just right, two scoops of rice, macaroni salad. The sweet potato haupia pie is excellent, too. Essentially no place to eat there (there was one small stool on the porch) so get it to go and head to the beach or a park to grind.
Little Fish Cafe (Hanapepe) - a new place for coffee and light food along the old town main street with a decidedly hippie style and really good coffee. I like the house coffee which they make fresh by the cup. For a great light breakfast I get their open-faced bagel sandwiches which come with a number of toppings; both the Hippie and the Veggie are very good. Menu covers a lot of blackboard space plus they have soups, banana bread, and more. Breakfast and lunch only.
Da Imu Hut (Hanapepe) - I don't think they have a sign: park in the Kauai Kookie parking lot and they are the place right alongside the highway. Hours can be variable, close early, possibly dinner only. The lau lau (pork, chicken) lunch plate is excellent and comes with lomi-lomi salmon. Very much not a fancy place but good food. Non-meat selection is extremely limited.
Obsessions (Waimea) - Another similar breakfast/lunch place in town that opens early so good to stop by on the way up to hike Koke'e. Surprisingly good Black Mountain brand (Hawaii) coffee for that kind of place where you get paper plates and plastic utensils.
Yumi's (Waimea) - Local home cooking by Yumi herself, breakfast and lunch (maybe some dinner hours). Outright the most friendly service on the island. Yumi also makes "goodies" - apple and coconut pies, pumpkin crunch bars, brownies - that a good size, delicious, and sell out quickly.
Island Tacos (Waimea) - I've become a regular here usually ordering Seared Wasabi Ahi fish tacos. The fish is raw but seared on the grill briefly, served with a wasabi mayonnaise sauce, purple and green cabbage, and white rice. They make their own tortilla for the soft fish tacos (and larger burritos) as well as fry them for chips. If you order just one taco you get chips on the side (two or more you have to buy the chips). I always get the hot sauce and in addition sprinkle some of the Hawaiian Habanero Heat really hot hot sauce they provide on as well for good measure. Limited hours are from lunchtime to five, seven days.