Why is the horizontal stabilizer root interface designed this way?
It is a seal- to prevent entry of air into the stabilizer mounting area. For aircraft with trimmable horizontal stabilizers (THS), there is an opening to the rear of the aft bulkhead to allow for the movement of the THS. It can be seen in the following image for Boeing 787.



Boeing 787 aft fuselage structure; image from aero-news.net

Now, as the opening will be there while the THS moves, a seal has to be present to prevent entry of air (and FoD etc.). Boeing went ahead and put a hinge aperture sealing plate over that area, with only a small opening where the actuated area of the THS moves, with the seal inside the structure, as can be seen below.



777 THS area; image from deicinginnovations.com

Without cover, the are will look something like this:



Trimmable horizontal stabilizer region without cover, ; image from skysoftairlines.blogspot.in

Airbus, on the other hand went ahead and put a tailplane sealing plate outside the opening. This was used in the A330 and initial versions of A340 (which shared the fuselage with A330), before Airbus changed its mind and made the THS region similar to the other aircraft in A340-500 and -600 versions, where the horizontal stabilizer was redesigned. 

Also, see these (1 &amp; 2) patents and this discussion.