What does "--" (double dash) mean in this shell command?

The -- tells cat not to try to parse what comes after it as command line options.

As an example, think of what would happen in the two cases if the variable $PIDFILE was defined as PIDFILE="--version". On my machine, they give the following results:

$ cat $PIDFILE
cat (GNU coreutils) 6.10
Copyright (C) 2008 Free Software Foundation, Inc.
License GPLv3+: GNU GPL version 3 or later <http://gnu.org/licenses/gpl.html>
This is free software: you are free to change and redistribute it.
There is NO WARRANTY, to the extent permitted by law.

Written by Torbjorn Granlund and Richard M. Stallman.

$ cat -- $PIDFILE
cat: --version: No such file or directory