That picture it looks like a high priced store to me. Personally for the most part I think that's bunk- unless you live in an area where you have limited stores and really are forced to shop at 1-2 places, most people should be able to save money eating healthier. That being said it's not EASIER to save money- I spend a LOT more time in the fridge, prepping food, making healthy meals, etc, but overall I believe I save at least $150 more a month- before I would EASILY spend $200 a week, now I spend more like $100 a week on groceries.
For one- you probably eat a lot less than you used to (I know I do!), another thing is that a lot of people don't LOOK around. You don't have to go to the chain stores- mom and pop grocery stores are still around, farmers markets are still around, and so on. Even in areas where fruit is more seasonal you can change it up and eat the foods in season to save. Heck if you are really struggling you can probably find programs in the area that will help you out- even get some food stamps for staples like milk and grains and so on.
Someone mentioned perishable foods go bad faster- my advice to that is buy smaller amounts of that food- go to the market a little more often if possible. Or take the time to do thing to make your veggies last longer. I will put paper towels in with my produce like romaine and anything that tends to get slimy from moisture- that helps SO much believe it or not. If you can't do that then buy veggies/fruits that last longer- apples, oranges, zucchini, and many other veggies and fruits last a lot longer than bananas, lettuce, and so on.
Also buying in bulk, you can buy a ton of veggies and stuff and then prepare them- once they are say in a large stew you can eat that stew for a few days and that way your food won't have gone bad either. I LOVE to make a pot of some sort of stew with my crockpot and then
Cooking lentils is SO easy- put your lentils in, a TON of water, some salt/pepper and any veggies you want, and set it on the stove, once it bowls turn it down and let it simmer a few hours (this is how I do it). Check on it periodically and add more water until the lentils are as soft as you want them