Consuming the same number of calories with different consumptions led to different amounts of calories being burned, and different amounts of weight being gained.
TLDR: what you eat matters, it's not just how much.
They are saying it in the technical sense that the people did in fact burn more calories on different diets. But those calories being burned were caused by the change in diet (assuming the hypothesis is correct). That is a massive change in American nutrition theory.
I'm around a fair number of fat people and I don't need any nutritional science background to notice they eat very often and they eat a lot. I suppose there's clinical evidence that low carb diets work, but it pretty much seems to me the main factor is these people just don't care about being fat slobs.
Also obesity has been an issue for middle aged wealthy people for at least the last few thousand years. Which suggest's people tend to eat more when they can afford to do so.