While iCloud Drive behaves like a full featured file manager on both Windows and Mac, that isn't the case on iPhone or iPad. There just isn't an iCloud Drive app on mobile. So, if you want to access iCloud Drive files on iOS, you'll need an App Store app to do it. The trouble is, not every App Store app supports every file type. So, if you just want to see all your iCloud Drive files on your iPhone or iPad, what can you do?
How to browse and manage iCloud Drive files on iOS with Documents 5
To view all your iCloud Drive files on iOS, I recommend Documents 5 by Readdle from the App Store. There are other apps that do it, like Goodreader, but Documents 5 has a friendlier user interface and is easier to get started with.
Launch Documents 5 (or Goodreader) on your iPhone or iPad.
Tap on the cloud icon in the bottom navigation.
Tap on iCloud Drive in the top left.
Navigate to the file you'd like to view and tap on it.
If Documents 5 supports the file directly, it will open. If not, you can always choose to open files in another app you have installed.
Where are you, iCloud Drive app?
In an ideal world, Apple would make an iCloud Drive app that serves as a repository for all our files, just like Photos serves as a repository for all our images and videos, and Passbook and Health serve as repositories for all our card and health data.
Note: iOS 8.1 (and OS X Yosemite) both seem to have a bug that sometimes causes the operating system to freeze when it tries to access iCloud. Hopefully this gets fixed in the next update.