Short version.
1. You would install osticket in a sub directory at your web host. (say at http://www.yourdomain.ext/contact-us(http://www.yourdomain.ext/contact-us))
2. You would figure out what file (or internal page) displays your "contact us" page and point it to the osticket directory. You will probably have to interface with support for your cart software to figure out how to do so, or utilize the opencart support forums. (available at their website)
They appear to have an integration section on their forums that you might want to try asking your question at.
http://forum.opencart.com/viewforum.php?f=128&sid=4f934fed5d76425aeb475ec74732d4a0(http://forum.opencart.com/viewforum.php?f=128&sid=4f934fed5d76425aeb475ec74732d4a0)