The app was initially restricted to European Xbox Ones but has now been made available to users worldwide.
Kodi's app isn't perfect just yet, as it was released in what is essentially alpha form. In a blog post, Kodi's Martijn Kaijser explained the limitations of the app:
What you should really understand and keep remembering is that it is still in early stages of development and has very rough edges, might not be as stable as the regular version and may even be missing some functions. Due to the nature of how UWP works our hands are tied in some areas. Some parts are not even finished yet and our developers are still working on getting it up to the regular standard. As of this writing, there's limited access to only what's part of your Video and Music folders.
Network support is limited to only NFS:// shares. No access to the Blu-ray drive to start the disc or even an attached storage drive. There might still be problems with certain general python modules that are used by add-ons and we are finding and reporting them to the developers as testing progresses. I'm sure there's more that might not work as intended yet as there are so many features it just will take a while to go over them. We cannot promise to what extent we can get every feature working as it all depends on what is available to us developers.
The bottom line is, you can now use Kodi on the Xbox One to stream your media as you would. Just don't expect it to be able to do everything you want yet, it'll get better over the next year. Kodi for the Xbox One can be downloaded from the Microsoft Store on the Xbox and is accessible via store search.