Go to mysms settings on your phone or the web app and click on "Dropbox, GDrive, Evernote, Email".
Connect your mysms account with the cloud service you like or choose to export your SMS text messages.
Depending on what you selected, all new messages will be forwarded to your preferred cloud service or a file of your entire SMS inbox will be created.
For Google Drive extension: Install this HTML viewer to view your text archive in Google Drive.
All plugins serve as an additional SMS storage. It is not possible to restore your texts back from your cloud service to mysms.