printf, scanf, and the other standard library functions are provided as part of the implementation.
A C implementation is made up of several components. The compiler is just one of them. The library is another; it consists of headers (commonly provided as source files like stdio.h) and some form of object code files containing the code that actually implements the library functions.
The header stdio.h only declares these functions; it doesn't define them. The declaration of printf is something like:
int printf(const char *format, ...);
The definition of printf is the code that actually does the job of parsing the format string, accessing the arguments, and sending the formatted output to stdout. That's typically (but not necessarily) written in C and provided as some kind of linkable object code.
For some C implementations, the compiler and the library are provided by the same organization. For others, they might be provided separately (for example MinGW combines the gcc compiler with Microsoft's library).