The C API, which looks to be the part that confused you, is designed to make it very easy for you to take existing C code (or new C code) and control it with a Lua script. This is what is meant by embedding. Via embedding, you can get a lightweight, programmable, interactive, text-based interface to any C code, for very little effort. Even if you don't embed things yourself, Lua is a very nice little scripting language.
My favorite example is that a nice person put a great many POSIX system calls into Lua. When I want something that is like a shell script but is more sophisticated, I can just use this library. Likewise with the MD5 library and with many others.
When learning Lua you can ignore the C API completely—just benefit from other people's work with the API—and if you ever need to embed your own code, you can come back to it later.
