Toward the bottom of your Private Messages page, you'll find something like below.
You'll have 3 format options; xml, csv, text.
XML is for easy import for certain applications.
CSV is for MS Excel, and OS X Numbers. Your PMs will be nicely organized in a table (columns and rows)
TEXT is all PMs in plain text.
You will download sent items AND PMs in your inbox.
Once you downloaded your PMs, you can then go ahead and delete to make your room in your inbox and sent items.