I think the white privilege that he was referencing was the ease and comfort of being blatantly racist, as well as shooting black people with no legal ramifications.
Now the story is odd, it might not be true. It may have been written to prove a point. But let’s shoe other foot it. Can an old black man talk to another random black repairman about how much he hates po trailer trash white crackers, cause they tried to rob him, and he shot them in the dick, and the police came over and laughed about the shot white guys? Is that a story that sounds feasible? Not really. I think that’s the white privilege, he might be referencing, but it’s an obtuse reference.