Here is the overview of the study:
It is interesting but there are a number of factors at play. There are also other studies that say diabetes and high blood glucose levels play a part in dementia risk factors. Also, we know certain fats are brain protective and if you aren't getting them then that can be a problem. The participants with the higher carb diet also reported eating a lot of sugar and ate few vegetables. So really the question here is the problem the lack of protective fats? Or is it the processed sugar that was part of the higher carb diet? The biggest limitation here though it is a self reported study.
Another thing is I see no mention of vitamin B-12 which is a known issue for seniors as absorption decreases as you age. Animal products contain B-12 so you'd expect seniors eating more protein would potentially have higher b-12 levels. Supplementation of B-12 is generally recommended for seniors, some more than others. Without Vitamin B-12, you can experience permanent cognitive degradation.
This is an interesting tidbit from the study as well:
Subjects with the highest % carbohydrate intake had the lowest total caloric intake which is consistent with the low % fat intake, but is also consistent with low BMI in these subjects, and with previously reported decreased weight loss in the years preceding onset of dementia in elderly persons [53–55]. In addition, moderate alcohol intake has been reported to reduce risk of cognitive impairment  and may play a role on MCI risk in our cohort. The dietary patterns observed may be causal or alternately, may be a marker for preclinical disease and risk of cognitive impairment or dementia in elderly persons. These associations need to be examined in other longitudinal studies.