Do you have remote desktop to a GUI environment on the Linux machine? If so, just log in that way and download the files with the web browser.
If not, using your browser on your local machine, identify the URL of the files you wish to download. Then open an ssh session and use one of the command line downloading tools to get the files directly on that machine. Some commonly-installed command line downloaders: curl, wget, GET (part of perl's www package).
Alternatively, you might want to use a terminal-based browser, like links, lynx or w3m.