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It's just conservation of energy. However this misses the point.

Sure, at the same level of exercise, if you reduce you caloric input, you will lose weight, regardless of the nature of the calories. If you don't, you won't. (This is because the calories listed on food labels take into account the energy that is actually usable by the body, and so for example doesn't count dietary fiber in the calories, which a simple bomb calorimeter measurement would count.)

However this is relevant only if you assume that humans are able to consciously control their caloric input. It has been proven time and time again that they can't. Not in the long run anyway, if they have a ready supply of food (which all of us who are not well below the poverty level do). Most humans can consciously control their caloric input for weeks, maybe a few months, but then fall back to a baser, subconscious control system for caloric input that relatively little is known about. Eating fewer calories is like trying to hold your breath, just over much longer time scales. A few people have the self-control to do this, but most don't.

The claim of some of these diets is that they manipulate this unknown control system in a way that permits lowering the set point of the caloric input in a way that can be maintained, without the human having to consciously consume fewer calories than the control system is requesting. It requires a different sort of conscious control of the food consumed, usually with regard to consuming some types over other types.

I suspect that most, probably all of the claims of these diets are bogus. However the basic idea has merit. If we understood the subconscious control system for caloric input, it would be far more effective to manipulate that system than to attempt, usually in futility, to override the subconscious demands with conscious control.

When we do learn about this control system, I'm sure it will be more complicated than just responding to total caloric input, so it may also be difficult to consciously control certain types of foods. (Do some "cravings" come from that system?) But however that system works, I bet that we can find a way to fool it into lowering the caloric input set point without drugs.

So while this study is correct, it proves nothing whatsoever about the possibility of diets that could in and of themselves help you reduce caloric input without having to count calories and force yourself to eat fewer of them. This study simply helped the participants to eat fewer of them with guidance and counseling over long time periods, with entirely predictable results. Also entirely predictable is that once the study is over, most of the participants will gain back the lost weight within a year.