Sheesh. You asked about this 9 weeks ago and no-body answered? Looks like I came to the wrong place for help.
I see this “View media” links under some people’s tweets now and I like them - you can read the first few lines of text from the blog, a wordpress blog, and you get the title of the blog and a link to the blog post.
So it’s a great feature and I want to use it. But there doesn’t seem to be any information in the twitter help center telling you how to do it.