This is pretty specific to PDFKit. I just took a look at the Middleware. It seems it is generating the PDF in-memory and sending it directly across to Rack. Check this:
https://github.com/pdfkit/pdfkit/blob/master/lib/pdfkit/middleware.rb#L11
If you need to be able to save them to your machine, you can write your own Middleware (you can copy-paste from pdfkit#middleware.rb, and customize it by adding File handling routines immediately after the PDF is generated - around line 21 in the above URL).
However, let me ask the dreaded question: why do you need to do this? Having PDF generated automatically for every URL the user visits does not seem to be a common use case. Unless you are playing around with it.
Let us know how it goes or if you need more help.