This is the journal article they're citing. The article you quoted is slightly incorrect in that the increase in waist circumference is what triples.
Adjusted for initial WC, demographic characteristics, physical activity, diabetes mellitus, and smoking, mean interval ΔWC of DS users (2.11 cm, 95% confidence interval (CI) = 1.45-2.76 cm) was almost triple that of nonusers (0.77 cm, 95% CI = 0.29-1.23 cm) (P < .001). Adjusted interval ΔWCs were 0.77 cm (95% CI = 0.29-1.23 cm) for nonusers, 1.76 cm (95% CI = 0.96-2.57 cm) for occasional users, and 3.04 cm (95% CI = 1.82-4.26 cm) for daily users (P = .002 for trend). This translates to ΔWCs of 0.80 inches for nonusers, 1.83 inches for occasional users, and 3.16 for daily users over the total SALSA follow-up. In subanalyses stratified for selected covariates, ΔWC point estimates were consistently higher in DS users.
Of course, correlation does not equal causation, so it may be that diet soda makes older people fat. It might be that fat old people drink more diet soda. It's also possible that there is some other factor or factors that both measures are influenced by. The article merely notes an association.
In a striking dose-response relationship, increasing DSI (Diet Soda Intake) was associated with escalating abdominal obesity, a potential pathway for cardiometabolic risk in this aging population.