There are a few ways you can save your files to your desktop from MATLAB Online and MATLAB Drive:
1) In MATLAB Online, you can download any file onto your desktop by clicking the "download" button on the toolstrip (under the "Home" tab). This will download the selected file in your Folder Browser to your desktop.
2) The other method is to download the MATLAB Drive Connectorhttp://www.mathworks.com/help/matlabdrive/ug/install-matlab-drive.html. This will create a MATLAB Drive directory on your local machine, and synchronize all the contents automatically with your MATLAB Drive/MATLAB Online. So, any changes you make to your files in any MATLAB Online session will be reflected in this directory (and vice versa).