It does sound a little confusing, though I think I understand your confusion (I could be wrong about that).
I think what is confusing you is what the "mirroring" is actually accomplishing, or you might be missing a step.
First, even though you are continuing to host your domain at your old host, make sure you have your domain added to your DreamHost account as a fully hosted domain. This is what creates the directory for the domain to live in (be served from) in your user's directory and sets up a DreamHost DNS record for that domain. Note that this DNS record is not used by the net at large as your domain is still set up to use your existing host's DNS record (not DreamHost's).
Once you have done that, upload your content into the directory that was created for your domain (usually yourdomainname.tld, though you may have changed it).
Now, if that directory could be "seen" by DNS, what was in that directory would be served to visitors - as that directory cannot yet be seen by DNS (because DNS is still being resolved by your existing host's nameservers, not DreamHost's), you can't get at it with a url using your domain name (DNS will point the user to your existing site on the other host).
- What the article is telling you to do is to add a "new" something.dreamhosters.com domain to your account, and set it to "mirror" your domain (the one you set up as "fully hosted" in step one - don't worry about DNS at this point).
Since DreamHost mirroring can only be used to mirror sites on DreamHost (maybe it helps to think of a "site" as a web accessible directory on a DreamHost server in this context), when you browse to the "something.dreamhosters.com" domain (address/url) that you set up for this purpose, the content that is displayed is taken from the directory on the DreamHost server that you set up when you added your domain.
As long as your DNS record at your registrar is still pointed to your existing host, visitors entering your domain name will see the content served from that host. This enables you to keep your site "live" over there, while making sure things are setup properly on DreamHost - you just put your content in the directory for your domain, but view it via the dreamhosters.com domain.
The reason this is particularly useful is that it makes the conversion "seamless". When you do change the DNS to point to DreamHost's nameservers, assuming you leave your content also available on your old host while updates are taking place, every user will see your site, whether the DNS servers they are using are updated yet or not. Those that have DNS updated will see your content served from DreamHost, and those that haven't will see the content served from the "old" host.
Once DNS has updated across the web, and all DNS servers are resolving your domain to the "DreamHost" site, you can dump the old one. Note that, unless you delete it, you will still be able to see the content also via the dreamhosters.com address.
Maybe part of the confusion is because of the way some other host/registrars use the term mirroring - some allow you to "wrap" content from another "off system" site in an invisible frame *appearing" to "mirror" another site and call that "mirroring".
If that kind of "mirroring" was what was taking place, you would see the "old" site being served from the "old" host", but that is not what DreamHost is doing. They are actually setting you up to have the content from DH "site" A actually served from the "domain/address" (directory on DreamHost) of DH site B.
I don't know if that attempt at simplifying the explanation of what is happening has made it better or worse, but I am hopeful it was helpful.
The DH Wiki article 'Mirror Domain" talks about some of this, and might help make it more clear what is actually happening.
You might also take a look at some of the "older" DH documentation on this, as it states it in sightly different terms than the wiki article: