Why can't someone import music from outside sources into apple's iPhone music app?
Apple calls the area in the storage of an iOS device containing media (music, video) for the built-in apps to play the iPod library.  iOS provides an interface that App Store apps can call to use the iPod library, but this is basically limited to playback and search.

Other than that, apps' access to storage is restricted by sandboxing.

From Apple's "iOS App Programming Guide":


  For security reasons, iOS places each app (including its preferences
  and data) in a sandbox at install time. A sandbox is a set of
  fine-grained controls that limit the app’s access to files,
  preferences, network resources, hardware, and so on. As part of the
  sandboxing process, the system installs each app in its own sandbox
  directory, which acts as the home for the app and its data.


This means that third party apps from the App Store can't directly access files other than their own (like the iPod library), so they can't save files that they download anywhere but their own directories.

In addition to the sandboxing, Apple also has a mandatory review and approval process for apps to go into the App Store, and so apps that don't follow the guidelines aren't available.  In a sense, that's not really a restriction "in the OS", because Apple can change their review process at any time and newly-allowed apps can be downloaded and used immediately without any change to iOS.  Also, iOS app developers can get their own apps signed to test on their own iOS devices, before the review process.  But from the point of view of the average user, what restrictions exactly are enforced by the sandboxing and what are only App Store guidelines is academic.  

On jailbroken devices, applications that have not been approved by Apple can be installed, and jailbreaks generally modify iOS so that apps can operate outside of sandbox restrictions. So it's possible for an app that understands the current iPod library file formats (for instance recent versions of iFile) to add music directly to the library.