The arguments to the interpreter in this case are the arguments constructed after interpretation of the shebang line, combining the shebang line with the script name and its command-line arguments.
Thus, an AWK script starting with
#! /usr/bin/awk -f named myscript and called as
./myscript file1 file2 results in the actual arguments
/usr/bin/awk -f ./myscript file1 file2 The single optional argument is -f in this case. Not all interpreters need one (see /bin/sh for example), and many systems only allow at most one (so your shebang line won’t work as you expect it to). The argument can contain spaces though; the whole content of the shebang line after the interpreter is passed as a single argument.
To experiment with shebang lines, you can use #! /bin/echo (although that doesn’t help distinguish arguments when there are spaces involved).
See How programs get run for a more detailed explanation of how shebang lines are processed (on Linux).