The point of the article is that drinking diet soda frequently messes with your internal connection between sweetness and being satisfied (full) in generally.
So in other words, people who drink diet soda will in general eat more and drink more of other sweet things in general and thus gain wait because their internal connections no long correspond to the real world.
That is why they would gain weight, not from the sodas directly.
It's not a bad theory. Interestingly, there is evidence of inputs into the brain reward system that operate independently of sweetness and that perhaps directly encode metabolic value: Rodents that lack the ability to taste sweetness prefer caloric sweeteners such as sucrose to noncaloric sweeteners, such as sucralose[1]. Therefore the brain can detect calories more directly.
So in the context of this new study, it could be that the brain notices that a certain rate of eating/drinking is correlated with a decreased rate of satiety and concludes that the correct thing to do is to eat more.
So in other words, people who drink diet soda will in general eat more and drink more of other sweet things in general and thus gain wait because their internal connections no long correspond to the real world.
That is why they would gain weight, not from the sodas directly.
Assuming of course that the hypothesis holds.