To start off , you have to be in the Discussion itself (Thank you, Dan Pihlaja). You can't attach files when replying to a Discussion from your Inbox. Now, if this is in a Reply to an existing Discussion, click on the black text "Use advanced editor" in the right corner above the text box. I know it doesn't look like a link, but it is.
After clicking on that you'll get a link to attach files in the lower right, beside the "Mention" link shown above. Click on the "Attach" link and Browse to and select the file.
If instead of a Reply this is a Discussion you're just starting, the "Attach" link should already be there. If you've already posted the Discussion and would like to go back and add a file to the original post there's an "Edit" link under "Actions" to the right of your original post you can click on to get the "Attach" link back and add the file to your original post.