There's no doubt that foods that are healthier are more costly in the short-term. Yes, absolutely, rice, pasta, potatoes and hot dogs are cheaper than eating fresh meats and veggies (or even frozen for that matter) so if you were used to a steady diet of boxed macaroni and cheese mix with maybe a can of corn, peas or mixed veggies on the side of course going low carb is going to increase your food budget!
On the other hand, if you were living a lot like we were, using a lot of pre-packaged convenience foods, you might find that your food costs go down once you bite the bullet and either learn to cook or start cooking your own meals once again instead of letting the factory do most of the work for you.
We're also buying a lot less expensive snack foods such as chips and expensive breakfast foods like cereals so breakfast and snacks are definitely more budget friendly around here now. Shopping wisely makes a huge difference as well. If the people who wrote this article were basing the cost on the sample menus in the books, I'm not surprised that they came up with the price tag that they did, but nearly all of those menus are impractical for the average person, both in time cost to prepare and the cost of the ingredients. We eat very well at our house, but not extravagantly. With a little advance planning, dinner rarely takes me more than 90 minutes to get on the table unless I'm making something special (and very little of that time is spent standing at the stove) and I cook enough so that there are leftovers for lunch; 2 meals for the price of one!
Lunch for me and my DH more often than not is leftovers that previously sat in the refrigerator in little plastic coffins until they began to growl at you every time you opened the refrigerator door. It wasn't too far from reality when my DH once purchased me a little magnet to put on the refrigerator that said, "If it crawls out of the refrigerator, let it go!".
One thing that the article doesn't address, though, is whether or not eating cheap highly processed foods is less expensive in the long run in terms of poorer health, loss of productivity/increased sick time from work and high medical costs to treat the diseases such eating habits cause.
When I look at it that way, eating like this is going to cost me much less in the long term even if my food costs may be a bit higher.