A couple of gotchas that are colloraries to Andrew Shelansky's excellent answer and to disagree a little with doesn't really change the way that the compiler reads the code
Because your function prototypes are compiled as C, you can't have overloading of the same function names with different parameters - that's one of the key features of the name mangling of the compiler. It is described as a linkage issue but that is not quite true - you will get errors from both the compiler and the linker.
The compiler errors will be if you try to use C++ features of prototype declaration such as overloading.
The linker errors will occur later because your function will appear to not be found, if you do not have the extern "C" wrapper around declarations and the header is included in a mixture of C and C++ source.
One reason to discourage people from using the compile C as C++ setting is because this means their source code is no longer portable. That setting is a project setting and so if a .c file is dropped into another project, it will not be compiled as c++. I would rather people take the time to rename file suffixes to .cpp.