Drafts support import of text files via iOS 8's Document Picker.
To access the document picker, tap and hold on the new draft button ("+"). A menu of options will appear. You can create a new draft, create a new draft already filled with the current contents of the system clipboard, or select from one of the document provider locations available on your system.
If you have iCloud signed in on the device, iCloud will be one of those options. You may also have additional document providers available through extensions in other apps installed on your device. Some apps which provide extensions to import documents include:
Transmit: allows direct access to FTP/Amazon S3 and other online accounts.
If you have not enabled these extensions, tap the "More" button to see what's available on your device.
Once you select a location, it's document picker will open and allow you selected a file to import.
To export files back to document picker destinations, read about the Export action step.
Drafts also suports pre-iOS 8 style "Open in" import. To import from apps that do not work with the document picker, look in those apps for "Open in", "Share" or "Export" buttons. These do not exist in all apps, but when they do, Drafts will be an option when opening text files.