Private variables are a way to hide complexity and implementation details to the user of a class. This is a rather nice feature. But I do not understand why in c++ we need to put them in the header of a class. I see two annoying downsides to this:


It clutters the header from the user
It force recompilation of all client libraries whenever the internals are modified


Is there a conceptual reason behind this requirement? Is it only to ease the work off the compiler?

It is because the C++ compiler must know the actual size of the class in order to allocate the right amount of memory at instantiation. And the size includes all members, also private ones.
One way to avoid this is using the Pimpl idiom, explained by Herb Sutter in his Guru of the Week series #24 and #28.
Update
Indeed, this (or more generally, the header / source file distinction and #includes) is a major hurdle in C++, inherited from C. Back in the days C++ C was created, there was no experience with large scale software development yet, where this starts to cause real problems. The lessons learned since then were heeded by designers of newer languages, but C++ is bound by backward compatibility requirements, making it really hard to address such a fundamental issue in the language.
