At the top of the message box you'll see a button that says "Upload via postimage.org". Just click it and follow the self-explanatory prompts.
Make sure you don't have "Web" selected instead of "Computer". If you don't see a "Browse..." button to upload a photo, then you have Web selected.
NOTE:If you choose to resize using the avatar or thumbnail choices the resulting BBCode is incorrect. Your image will show up in your post, but when it is clicked on the upload image dialog will appear instead of the your image. Therefore it is highly recommended you do not use these size options.