What are '.save' files?

It is a nano emergency file where its buffer gets dumped. If your current file seems alright, you can delete the filename.save associated file.

From nano manual notes:

In some cases nano will try to dump the buffer into an emergency file. This will happen mainly if nano receives a SIGHUP or SIGTERM or runs out of memory. It will write the buffer into a file named nano.save if the buffer didn’t have a name already, or will add a ".save" suffix to the current filename. If an emergency file with that name already exists in the current directory, it will add ".save" plus a number (e.g. ".save.1") to the current filename in order to make it unique. In multibuffer mode, nano will write all the open buffers to their respective emergency files.

Are you using the Nano text editor? If so, this two links might help

The save file extension is associated with the command line text editor Nano that is a part of various Unix and Unix-based operating systems. The *.save file contains temporary saved text file. It is backup used to protect text document against system crashes. After the text is saved by user the temporary save file can be deleted.
reference: www.file-extensions.org

In some cases nano will try to dump the buffer into an emergency file. This will happen mainly if nano receives a SIGHUP or SIGTERM or runs out of memory. It will write the buffer into a file named nano.save if the buffer didn’t have a name already, or will add a ".save" suffix to the current filename. If an emergency file with that name already exists in the current directory, it will add ".save" plus a number (e.g. ".save.1") to the current filename in order to make it unique. In multibuffer mode, nano will write all the open buffers to their respective emergency files.
reference: nano manual