So with that, it keeps the data(state) in the public facing class. This allows one to keep everythign stack allocated instead of defaulting to the heap. So for something this is created en mass(A vector of them) or created and destroyed often, this is a runtime win.
What it does is have a proxy that mirrors the public interface that is passed the this pointer. That proxy a friend class. Because only the proxies header(in this case private.h which I should probably rename firewalled.h) has static members that mirror the public members on the public class that limits the interaction between your classes users and it's implementation, as it is also with unique_ptr(or whatever pointer/heap way) based PIMPL designs. So changes in private.cpp that does all the work are only reflected in that one file. This file also brings in the heavy templates or algorithm code that may have large compile times.
