Why would I use the MySQL "system" command? What can I do with the MySQL client's system command?  Why does it exist?

The most productive use I've seen is to look around the file system (e.g., to remember the file name you want to SOURCE or LOAD FILE).

mysql&gt; system ls /tmp
backup.sql 
mysql&gt; source /tmp/backup.sql
Query OK, 1 row affected (0.02 sec)

mysql&gt;


As near as I can tell, you can't pass a query result to the STDIN of a command, you can't pass STDOUT of a command into a MySQL query.. it just doesn't seem widely useful.
There is actually a far more useful application for the SYSTEM command than simply executing ls to see the current directory contents:

As mentioned in the docs (and elaborated on by a comment), this is way to make backups:

echo "FLUSH TABLES WITH READ LOCK; SYSTEM snapshot.sh; UNLOCK TABLES;" | mysql


snapshot.sh contains code to make an atomic snapshot of the mysql tables, using LVM, ZFS or btrfs capabilites.
You can't split this up into multiple commands from the shell script, because as soon as the mysql session is closed, the table lock is released.