Apple has always kept file storage hidden away on the iPhone and iPad, but that's changing with iOS 9. Thanks to a hidden app you'll be able to see the folders on your iCloud Drive— much like you can in the Mac's Finder—on your iPhone and iPad.
iOS 9 includes an app to view what's on your iCloud Drive
The iCloud Drive app lets you view the contents of the folders on your iCloud Drive account, plus you can tap documents to open them in the apps that created them. You can't, however, create new folders in the iCloud Drive app. For that, you'll still have to head over to your Mac.
To see the iCloud Drive app you first need to have iOS 9 installed on your iPhone or iPad. That's currently available only to developers, but come July anyone can try out the public beta. The official release is coming this fall as a free update.
Once iOS 9 is installed, just go to Settings > iCloud > iCloud Drive and tap Show on Home Screen. On a side note, I'd love for Apple to add this setting to other default apps so we can hide the ones we don't use instead of tucking them away in an unused app folder (mine is called Rura Penthe).
The iCloud Drive app in iOS 9 is hidden by default
This isn't full-on file management on our iPhones, but it's still a big change on Apple's part. Before iOS 9 there wasn't any way to see or manage files on our iPhones and iPads, so maybe this is a sign that Apple is seeing mobile devices as more of an alternative than it did before for on the go computer users.