What that code snippet is doing is accepting a status from its caller and transmitting it back to whomever actually started the application and exiting the application. Say my program wanted to open a file "foobar.dat" and failed to open it. I would take the status given by the Fortran Open command (iostat=my_open_fail_status) and call EXIT_WITH_STATUS(my_open_fail_status). It would then cause my program to exit and return "my_open_fail_status" to whatever had actually called the program. That could be the dos shell, or in Unix, whatever had spawned the program.
If you are using the Simply Fortran GCC compiler, you can replace EXIT_WITH_STATUS(me_status) with CALL EXIT(me_status) and everything will work identically. "Call Exit()" is a language extension and not part of the standard as far as I know, so you may wish to use the "error stop" construct such as "error stop me_status, quiet=.true." to stop the program as you can use that out of a pure subroutine, and no stderror output but pass the exit status to the invoker.
I hope this helps and is clear enough. If not, reply and & try to clarify]]>