I think of the half halt as momentarily compressing the horse like a bow. (meaning bow and arrow) By asking for more impulsion from behind while shutting the door in front, the energy has nowhere to go but up, hence the raised back aka bow.
As to the mechanics, it's become more of an overall complete movement for me, but if I had to break it into parts, I'd say stilling my seat and closing my hand on the outside rein pauses the legs front legs. At the same time asking for more impulsion by closing my legs, brings the hind legs forward at that same time. The result being the hind legs are now more under the body since the front legs didn't go as far due to the halted stride. Again back has nowhere to go but up.
You just have to see your distance...you don't have to like it.