As I recall, they didn't really figure out the reason for that, but the paper suggested one potential problem is that adding lights simultaneously ruins people's dark vision and creates more shadows to hide in. One other hypothesis I'd want to rule out before drawing any big conclusions is that law-abiding citizens aren't the only ones who know that people choose the more brightly lit streets when walking at night.
My own personal best guess is that, if there's any relationship, it's non-causative. I'm guessing that what muggers actually prefer is areas with low foot traffic, and it just happens to be that places with less foot traffic also tend to have fewer lights.
In contrast, the college student slum near my university had high foot traffic. From my biased sample, I estimated that 1/10 student residents of that neighborhood were mugged in a 4-year period.
Also, a street lit well in some pockets but not others ruins night vision and creates hiding places. A residential street with lights on at every door might look nice at night but may also be worse for safety. Except for the psychological deterrence of "people live here, are home, and care enough to leave the light on."