Use the entire VIDEO_TS file. Don't just use one of the files within VIDEO_TS, as can be seen in the screenshot you included along with your post.
Just pull up VIDEO_TS in HandBrake, adjust any settings depending on your purpose for encoding (i.e. to watch it on you iTouch or iPhone), and you're good to go!
And no you dont have to manually encode all of those files under the VIDEO_TS file, just do VIDEO_TS as a whole