The participants received each diet type for 4 weeks, and their resting energy expenditure (REE - the amount of calories your body needs to burn just to keep you alive) and total energy expenditure (TEE - the amount of calories you burn for everything you do) were measured. If you're trying to lose weight you would hope that your REE and TEE are high, because the more calories you burn the less fat you're going to be walking around with. You can see from the graph below that there was a slight decrease in REE and more substantial decrease in TEE when the participants ate the low fat, high carbohydrate diet relative to the high protein, low carbohydrate, Atkins like diet.
The authors of the study go on to say that they can't explain this change in TEE by an increase in metabolism caused by changes in thyroid hormone concentrations, or by the amount of physical activity each participant had. They hypothesize that a low carbohydrate diet may increase your TEE by changing other hormones in the body, or that the differences in metabolism of proteins, fats and carbohydrates may be responsible. Dr. Hirsch disagrees, stating that this TEE effect is probably due to the fact that when you are on a low carbohydrate diet you lose a lot of your water stores, and because lean body mass includes those water stores, your lean body mass will actually decrease. Because TEE is usually calculated as calories per unit lean body mass, you can calculate an artificial increase in TEE when the participant's lean body mass goes down.
I have to say that I like Dr. Hirsch's perspective on this, not because I'm strongly against a low carbohydrate diet (just strongly against a high meat one), but because he ultimately concludes that if you want to lose weight, you should just eat less of whatever it is you're eating already. As exemplified by the Twinkie diet, this works. You can eat Twinkies all day if you want and still lose weight, as long as you only take a micro-bite of the Twinkie and stop the instant you're no longer starving.
Ebbeling, CB et al. Effect of Dietary Composition on Energy Expenditure During Weight-Loss Maintenance. Journal of the American Medical Association (2012)