There are two ways: When you begin a new message, you'll note there are lots of icons above the message area. To the right of the "smiley" is a paper clip icon. Click on that, and you'll get a pop-up box where you can upload your attachments. All attachments will appear as thumbnails if they're over a certain size.
The other way is to look below the message area under the Additional Options. The second box there has a button that says "Manage Attachment" and will open the same box as the paper clip icon does.
If you're editing a message and want to add attachments to it, you'll see a button right next to the Save button that says "Go Advanced." If you click that, you'll get a format that looks just like when you're posting a new message.
If you use the Quick Reply feature, you'll see a "Go Advanced" button there, too, which serves the same pupose as the one in the edit screen.