But isn't that what this is? I was under the impression that a compiler turns source code into machine code whereas a transpiler turns source code into differnet source code?
All compilers turn one form of code into another. Whether that's C to assembly (to machine code), or JavaScript to bytecode to machine code, or Python to Python. Transpiler is just a term referring to a compiler taking one high-level language to another. It's unnecessary because a Coffeescript to JavaScript compiler is a transpiler already; saying "transpiler" there tells you nothing you don't already know.
Compilers are computer programs that translate a program written in one language into a program written in another language.
Example of a compiler in action.