When you arrive at the draw.io application web page, you'll see the storage splash screen


 

Whichever mode you select, if you tick the check "Remember this setting" (and have cookies enabled) your choice will be remembered and returning to the draw.io web site will skip the storage selecting splash and start with the new/open splash instead.
 
The 3 options (usually) available are:
  1. Use your Google Drive account. See Using draw.io with Google Drive for more details.

  2. Use your Dropbox account See Using draw.io with Dropbox for more details.

  3. Save to your local filesystem (not available on tablets and mobiles, for example). Using draw.io with your local file system is very simple, but also limited in a number of ways. Autosave is not possible, nor is it possible to define the save to directory on the file system, these are security limitation enforced by all mainstream browsers.


When opening from local storage this simple dialog appears:
Allowing opening of diagrams by clicking on the link and diagram deletion by clicking on the corresponding trash icon.
Note that local storage changes the URL so that refreshing the page keeps draw.io in local storage mode using the same diagram, e.g. https://drive.draw.io/#Ltest2. To switch away from local storage mode you must alter the URL directly.