The "plugin" turns out to be a .sh file that you need to make executable, and then run:
sudo chmod a+x aspera-connect-126.96.36.199118-linux-32.sh ./aspera-connect-188.8.131.52118-linux-32.sh
This installs a plugin for firefox which is available when you close and restart your browser. I don't usually use firefox but I find that usually even my enterprisey clients have heard of it and will sometimes support it! At this point, I could download the file and got a little progress window. I left the file downloading overnight.
This morning, I came into the office and looked for my file. Since Firefox insists on dumping downloads all over my home directory usually, I expected it to be there, but it wasn't. It turns out that the aspera faspex client will download the file into
/tmp on my ubuntu system (actually Linux Mint based on ubuntu 11.10). Since my
/tmp gets wiped every time my machine reboots, I moved the file fairly quickly! Hopefully this helps anyone else trying to use the same combination of tools.