They are part of C. A C implementation consists of a compiler and a library (and other components, like a linker).
The C core language includes facilities that make it possible to write library code that can be used by other programs. The standard library portion of the standard specifies a library that can be implemented using the facilities defined in the core language.
Some languages do have things like a print
command built into the language. C's facilities for writing and invoking library code written in C are powerful enough that that's not necessary.
Furthermore, most of the library is optional for "freestanding" implementations (mostly for embedded systems). There are implementations that support the full core language but that don't provide most of the C standard library.
And a compiler and library can be provided separately. For example, gcc is a compiler; it's commonly used with different library implementations on different systems (GNU libc on Linux, "newlib" on Cygwin, the Microsoft library on Windows with MinGW, and so forth). Mixing and matching like that would be much more difficult if the library were integrated into the core language.
The C language standard (the link is to the newest freely available draft) defines C. Section 6 defines the core language; section 7 defines the standard library.