If you look at previous research on exercise, inflammation, and brain volume:

http://dx.doi.org/10.1016/j.neuroscience.2014.05.005

and the mountain of evidence showing that excess visceral fat tissue produces greater levels of chronic inflammation:

http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3437966/

Then it seems that inflammation would be the first place to start looking for mechanisms linking weight with white matter volume.

Certainly more visceral fat is associated with higher risk of all of the common forms of dementia, and most age-related conditions are sped up in their progression by inflammation.