Two great, clear explanations here. The only thing I would add is that the half-halt doesn't last very long. It only takes a second. It's often invisible to someone watching, but you sure can feel it from the back. You can feel the horse gathering its power underneath you, then when you release the half-halt and give the cue for the next thing, that power goes into whatever you're asking. The first time you do it correctly, it will be very obvious to you (light-bulb moment).
Are we allowed to link to youtube videos that aren't our own? I'll take a chance:
This video helped a friend of mine understand it better.