To start off, you have to be in the Discussion itself. You can't attach files when replying to a Discussion from your Inbox (thank you Dan Pihlaja for pointing that out).
If you're attaching a Drawing or Assembly please keep in mind that you also have to attach all referenced files or we won't be able to open it, and don't post these files separately, but do a Pack and Go to a zip file and post that (see How can I create a new Assembly or Drawing similar to an existing one? if you aren't familiar with Pack and Go). If you're posting a Part file it's unlikely that you need to zip it first. Just post it directly. And don't attach .rar files. Very few people will open them. I know I won't.
Now that I have that out of the way, 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 left corner. 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, click on the "Edit" link near the top right of your original post to get the "Attach" link back.
2018-09-12 edit: If you only want to attach a screenshot you also need to use the advanced editor to post a jpeg or similar file, but there's an easier way. My usual procedure is to hit the PrtScn/SyRq button on my keyboard to copy whatever's on my monitor. Next I open Paint and hit Ctrl + V to paste the screenshot. I use the tools in Paint to underline, point an arrow at something, etc, then use the Select tool to choose a specific area. Ctrl + C to copy it, then go to the open forum post and Ctrl + V to paste it directly in with the text. It's that simple. Other users have somewhat different workflows that I'm sure work just as well (and they're free to post theirs here if they'd like), but this one works well for me.