Round here we always say we work out for our mental health as much as our physical health. If you move regularly you can probably relate to the concept. It always sounds glib, but it turns out we may actually be onto something. 

Across multiple studies it seem that overall muscle strength is consistently linked to better cognition and reduced risk of dementia. We say 'overall' as most studies concentrate on one aspect, some leg strength, some grip the some try to get an overall level. The reasons for specifics such as grip or leg are usually prosaic. It's much easier, and safer, to just have someone grab a grip meter or sit on a leg curl machine than to expect them to pull off a compound lift.  Regardless of the muscle group or movement though, they all come to a similar conclusion. Strength seems to predict cognitive health as we age.

There are caveats though (aren't there always). It's not proven to be a direct cause. For example strength may be linked to better overall fitness therefore making it more likely stronger people eat healthily or make other good choices. Interestingly, one of the studies is a twin study so they manage to rule out genetics as the influence.

So we're not sure exactly why the relation is there but it's consistent enough you should take note. In time it may turn out that increased power is more important or that working on your strength is linked to decreased visceral fat, or some other mechanism, but the fact remains that a good way to protect your precious grey matter is to get out there and move. 

Pro tip: Try and enjoy it. It's much easier, and stickier, if you do.

Studies:
Leg strength and cognitive health in twins.
Grip strength and cognitive health.
Muscle strength and Alzheimer's.