ldconfig takes all the shared libraries you have on your system, and caches them in a large file. (for files to be indexed their path must be present in /etc/ld.so.conf, the file they are cached to is ld.so.cache)
This also creates symmlinks. So as before say you had libsomething9.so
ldconfig will know what you want, and for compatibility will also create, libsomething.so libsomething.9.so and a few others.
This tends to fix most problems people have with libraries. Most libraries look for simply the libsomething.so, so you need ldconfig to make that symmlink. If not you can make it by hand, but it's always better to let ldconfig handle it. It's better at it
compat-libstdc++ is jus t what it sounds. libstdc++ is the standard library for C++. This library is a compatibility library, so you can run programs that use functions which aren't available in libstd from linux.