OK here's the deal. Sign in to flickr and click the "Upload" link roughly centered at the top of any page. Select the photo or photos you want to share from your computer, name and tag them as you wish, that part is pretty self explanatory. Once you have images in your flickr stream you have to click on one to go to the photo page, just like always. Now at the bottom right of the huge black portion around your image you'll see some icons.
Click the rectangle with the arrow swooping out of it for the share options. This window pops up. Hit the code tab as shown above, make sure BBCode is selected, and pick an image size. Then you copy that code and paste it into the reply/post form here. Still pretty straight forward.