Oddly, the kid's letter didn't even use the term "white privilege" -- that racial term was added by the political website that was linked here.
I think this kid widly misunderstands what his fellow students are saying to him when they say "check your privilege." He is privileged if he comes from money -- as he apparently does. This term gets used when kids flaunt attitude that they themselves didn't earn. Yeah, his family worked hard, but he didn't. (I'm pretty sure he would've mentioned it if he had ever done any work himself.) Probably nobody ever told his parents or grandparents to "check your privilege." I'm guessing he behaves like an entitled ass, and his peers are actually doing a him a favor by pointing this out to him.