The two groups

BBT (Body-Brain) - Trained in person on a physically interactive custom video game specifically designed to be challenging on a number of different cognitive skills, as well as physically demanding (relatively...for old people). From the paper "requires participants to perform full-body, physical movements in response to cognitive challenges that engage different cognitive control domains. As in our previous work, we integrated real-time adaptivity using closed-loop mechanics for each cognitive ability being challenged. Given the movement demands of BBT, we also implemented an adaptive physical fitness challenge by using real-time heart rate data to titrate gameplay for a personalized and integrated training experience across both cognitive and physical domains"

MBT (Mind-Body) - Trained remotely on a tablet using some combination of 3 different commercially available apps. The apps were chosen because (essentially) the researchers believed participants would expect the apps to improve their cognitive performance. Research team put in some unspecified effort to have participants journal their daily app training and to support participants rotating among all 3 apps. Specifically told them to practice sitting down (sedentary). From the study: " For those participants randomized to the MBT group, they were loaned an iPad tablet (9.7 inch screen size; 1024 × 768 screen resolution) for their training session following their ‘Pre-training’ assessment, and were instructed to train with their assigned task at home for six weeks, 5 days per week, with 30-minute training sessions per day, for a maximum of thirty 30-minute training sessions (15 hours of training total). All MBT participants were instructed to train sitting down with the tablet on a flat surface, such as a table, in a location with minimal external distraction."

>And the study outcome was “increased attention”. Quite often similar studies have found that they’ve trained a group to perform a specific task well, but not globally increased intelligence, attention span, etc

I would have to basically quote the whole methods section to explain, but the simple response is this: The goal of the study was to show their intervention had a measurably larger effect on target cognitive skills than a placebo intervention. They planned the study and the intervention based on the specific target cognitive skills of interest. It's well known that interventions need to 'train up' specific cognitive functions. In fact, they included pre- and post-testing in cognitive functions they expected to see no difference in to support their hypothesis that this intervention works the way they expected.