In order to download your own papers, you will need to select the dropbox for the paper. This will take you to the file submission that you uploaded and will give you the option to download this file.
When you log into Blackboard, you can access your courses and any course files (usually in "Course Documents" or "E-reserves"), as well as any papers you submitted on Blackboard (in "Final Paper Dropbox" or other file submission assignment dropboxes). When you click on any pdf course document or reading that your instructor uploaded, you will see something like the image below once the pdf opens in a new window. The button with the arrow pointing down will allow you to download this file.