From Kass Yassin, the head guy here. Hope this answers your question as the box you're asking about is the blurt box.
Rooster Cogburn. 6/10/2015
Kass Yassin blurted 5/24/2015
What does it mean to Blurt an answer?
When you "Blurt" a question or answer, it shares that page with your followers on Blurtit.
It will appear in your follower's feeds as: "your name) Blurted..." and then shows a preview of the question/answer page.
The concept is similar to a "re-Tweet" on Twitter.