What does `kill -0 $pid` in a shell script do?

sending the signal 0 to a given PID just checks if any process with the given PID is running and you have the permission to send a signal to it.

For more information see the following manpages:

kill(1)

$ man 1 kill ... If sig is 0, then no signal is sent, but error checking is still performed. ...

kill(2)

$ man 2 kill ... If sig is 0, then no signal is sent, but error checking is still performed; this 
can be used to check for the existence of a process ID or process group ID. ...

/>
static int
checkRunningPid(void)
{
    pid_t pid;
    debug_log = stderr;
    if (strcmp(Config.pidFilename, "none") == 0) {
    debug(0, 1) ("No pid_filename specified. Trusting you know what you are doing.\n");
    return 0;
    }
    pid = readPidFile();
    if (pid < 2)
    return 0;
    if (kill(pid, 0) < 0)
    return 0;
    debug(0, 0) ("xxxx is already running!  Process ID %ld\n", (long int) pid);
    return 1;
}