Can you explain how this makes sense?
Different physical activities (running vs. sitting, for example) require different amounts of energy to perform. Calories are a unit of measurement for this energy. It sounds like you (via referencing the article) are claiming that a human will expend the same amount of energy in a given period of time regardless of their physical activity.
Yes, that's exactly what the article is saying. If you burn calories doing exercise, the body will spend less on stress response. But if you burn less (or none) on exercise, then it will spend more on other things, eventually coming to roughly the same calorie expenditure (as they measured). Which is why exercise helps with depression - your brain has less calories available to spend on stress responses and generally messing with your mood.
You really should read the article.
And yes, it allocates calories towards building fat. Which is a weeeee different from burning them.
tl;dr: your body burns roughly the same amount of calories regardless of exercise. However, it allocates those calories differently: if you don't spend them on exercise then it will spend them on stress (and presumably other stuff).
Which totally makes sense as to why my depression & anxiety fade if I run or work hard.