Usually the installation has to be confirmed manually, with the iPhone telling the user who it's from and whether or not it's a trusted source. However, the hackers were not only able to make the configuration file report back as "Verified," but also indicate that it was from "Apple Computer” themselves.
Once installed, the file could allow hackers to redirect all traffic through a server of their choosing. The file could also disable the use of Safari, Mail, and a handful of other first-party iPhone apps. Moreover, it's possible to set the configuration file so that the user can't remove it without a full wipe.
You can find the full technical details here.