Although it's been awhile since I've read the book, I think the Eadses recognize that even though burning fat for fuel has a metabolic advantage, you can't go hog wild on it. You still have to maintain a caloric deficit in order to continue losing weight, especially after the first 50% of your weight loss. Even if you don't pay attention to calories at first, eventually, you have to.
Of course, if you exercise, you get a break, because your caloric expenditure goes up. If you do weight training, your increased muscle volume uses up some more calories, even at rest. But there's a limit.
Once you've settled into a LC routine, take a look at any of the BMR
calculators on the web to determine what your caloric intake should be to maintain your weight. About 10% below maintenance intake is all you need to aim for to continue losing weight. Going too low below maintenance intake will signal your body that you're starving, and you probably won't lose weight then.
It's easy to get calories from fat; 2 oz. of pure fat has about 450 calories (9C/g x 25.4 g/oz). Many LC'ers, myself included, get 50-65% of our calories from fat. The difficulty is finding healthy sources of fat. I'm lucky in that many local organic ranchers who sell at farmers' markets will sell their animals fat, which I then render.