Regarding your general questions:
Why the C library works:
The C library is part of your cross toolchain. That’s why the headers are found and the program correctly links and runs. This is also true for some other very basic system libraries like libm and libstdc++ (not in every case, depends on the toolchain configuration).
In general when dealing with cross-development you need some way to get your desired libraries cross-compiled. Using binaries in this case is very rare. That is, especially with ARM hardware, because there are so many different configurations and often everything is stripped down much in different ways. That’s why binaries are not very much binary compatible between different devices and Linux configurations.
If you’re running Ubuntu on the Raspberry Pi then there is a chance that you may find a suitable ncurses library on the internet or even in some Ubuntu apt repository. The typical way, however, will be to cross compile the library with the specific toolchain you have got.
In cases when a lot and complex libraries need to be cross-compiled there are solutions that make life a bit easier like buildroot or ptxdist. These programs build complete Linux kernels and root file systems for embedded devices.
In your case, however, as long as you only want ncurses you can compile the source code yourself. You just need to download the sources, run configure while specifying your toolchain using the --host option. The --prefix option will choose the installation directory. After running make and make install, considering everything went fine, you will have got a set of headers and the ARM-compiled library for your application to link against.
Regarding cross compilation you will surely find loads of information on the internet and maybe ncurses has got some pointers in its shipped documentation, too.