Studio 13.37 saves all of your changes - whether just the desktop background, or adding files you downloaded off the Internet - to a file called studiosave.sfs. If this gets too full, you will have to copy to another drive and resize it, format it using Factory Reset, or use the "boot command" method (bottom of post).
The "storage" partition, called sdx
2 on the desktop, is a place you can keep your files outside of the system, where they can be accessed on any computer without having to boot Studio 13.37. Placing files here can keep your savefile from getting too full. Transferring files you aren't using to a home computer is another good idea.
And if you find yourself using Studio 13.37 as your main "studio of choice", you might want to consider installing it to your computer's hard drive. (Here's a link
to a video on how to do it.) This removes the need for a savefile altogther.
"Boot command" method
If you have to, you can type "puppy pfix=ram" at the boot screen with the Studio 13.37 logo. This will load Studio 13.37 without a savefile, and allow you to delete the old one, or remove files from it by clicking studiosave.4fs and then moving the files. The latter allows you to free space from your savefile without deleting or formatting it.