Question: I understand and very much appreciate the practice of separating header (.h, .hpp) and source files (.cpp). 

However, many simple classes can be expressed entirely in the header file itself. I could define some methods within the class and others (e.g. template methods, inline methods) below the class. 

What advantage is there to creating a .cpp file when I could just declare *and* define the entire class in the header file? (Do I even need to?)

Answer: 
If you have the code in the header file, whenever the header changes, the client code that includes the header must also be recompiled. With a separate cpp file, only re-linking is needed. On large projects, the difference may be in hours.

Putting the code in the .cpp file hides it from the user, if you build a library and distribute only the header file. Sometimes this is necessary.

By having a header file, you can leave the client code unchanged and replace the library (say between debug and release, or regular and call-tracing versions). This is useful for debugging or investigating problems.

Leaving the header file aside makes for a smaller file that is easier to understand by the class user. You can hide complexity away in the .cpp file.
As already mentioned compilation time is one reason for having separate translation units. Especially if you have many cores parallelization can dramatically decrease compile time. Another reason is that non-template classes and functions as well as variables in header files would lead to duplicate symbols when included in different translation units. As soon as you use virtual functions inlining is no viable option.

You could use templates to avoid the duplicate symbol issue but in addition to higher compilation times this also leads to higher linking time because the linker has to collapse the template-instances (see <https://gcc.gnu.org/onlinedocs/gcc/Template-Instantiation.html>). Nevertheless STL and many other header-only libraries do exactly this.