WK was having a bit of fun here. I don't know if he was trying to see if any Americans would rise to the bait or didn't realise the joke wouldn't necessarily travel across the Atlantic.
For historical reasons (i.e. it makes no sense but it's the way it is) in Britain the term "public school" actually refers to a private school - usually also a boarding school - and they are often regarded as being only for the rich upper classes.
Hence the "Be off with you or I'll have my man thrash you and set the hounds on you" - a stereotypical representation of the sort of thing an upper class twit might say to you if you suggested abolishing the fee-paying public schools his family have been packing their children off to for centuries.
"my man" tells you he has servants and "the hounds" means he is probably a landowner who keeps a pack of hounds for hunting foxes with.
Oh and giving thrashings to the peasants was a common practice until recently!
I promise you his post was funny over here
I wish I didn't know now what I didn't know then