First of all, this app was initially designed without smartphones (touch devices) in mind. But, you can open and view your mindmaps and do modifications. We are soon implementing support for touch friendly devices.
You can open maps stored in google drive.
To open it via browser in smartphone, point it to http://app.mindmapmaker.org and click on 'Document'->Open->'Open' beneath 'Google Drive'. There you can choose existing maps like the way it is done via desktop.
To open it via, 'Drive' android app, click on the map file. If it says the filetype is not recognized, then you haven't associated 'mindmaps app' with JSON filetype. First associate 'JSON' filetypes with 'mindmaps app' and then you can open drive files from 'Drive' app directly.