What they're saying is that when the encrypted partition is mounted, access to the data is controlled by whatever means you would normally use (and not by the encryption itself). For example, you can have an encrypted
/var/database for your HTTP/SQL server, but if you get exploited online (think Internet) then the encryption isn't standing in the way of the bad guys, your other security measures (ACLs, MAC, etc.) are. The HTTP server normally has access to the database, and now that the bad guys have taken over the HTTP server (think arbitrary code execution), they can do whatever the HTTP server can do.
The link you shared goes on to explain that if you need certain data to be protected by encryption during an online attack (or a local one, what's important here is that the computer is up and running and the encrypted partition is mounted), you'll need to use file-based encryption. This way, the key isn't loaded and the data isn't being encrypted/decrypted on the fly, protecting it even if your other security measures fail. For example, if I leave an AES 256-encrypted gzipped-tarball on the server, say in
/usr/local/backup.tar.gz.aes, it doesn't matter how bad the server gets compromised, as the bad guys would still need to get their hands on the key in order to access the data in the clear.
As for the security which partition/disk encryption actually provides: A proper shutdown stops usage of the partition/disk encryption key (and ideally, overwrites it in RAM, so that it can't be retrieved later via
cold boot attack, but that's another story), protecting the data even if the computer or its storage device is physically stolen.