By choosing the option to have us change your nameservers for you (we know it can all get very confusing!), all we need from you is your Domain Key, or Domain Password, that you would have been provided with when you originally registered your domain.
When you register a domain name, part of the process is setting the DNS (Domain Name Servers) for your domain. They quite often look like "ns1.yourhost.com" and "ns2.yourhost.com". They are basically a directional marker, telling a user's browser where to go looking for the website the user has typed in.
If you have an existing domain name, the DNS entries could be set to your Domain Registrar's name servers, or to your previous web hosting provider. Because you are changing your web hosting to Brisbane City Web Host, the DNS entries need to be updated to show our name servers, so that everyone can find your website hosted on our servers.