They work on the simple principle that if you control the snout end of the dog (the head), you can gain control over the rest of the body. Exactly like a halter for a horse.
They can be a wonderful way to keep a hound's over-active nose off the ground, so you actually can get her focus enough to realize what you are asking.
They can be a fantastic way to teach a puppy how to follow you (there are some assistance dog organizations that have moved away from using choke chains in favor of these), and on the right dog, they can be a valuable tool to help break the focus of a reactive dog, and get them on the road to non-reaction.
Here again, is a Tool that can easily be misused.
This is not a tool for a dog that launches himself to the end of the lead with such force that he sometimes either yanks you down, or flips himself over. The head-halter attaches to the dog's skull, and the skull is attached to the vertebrae of the neck. A dog moving at high velocity to the end of the leash in a head-halter, is at risk of snapping it's neck.
This is not a tool designed to be "popped" or "yanked" by the human. I see Humans all the time yanking quickly and snapping the head of the dog in a twisting motion, as they say "NO!" or "Leave IT!" You can hurt your dog.
Don't do this.
Wrapping the leash around your hand and using an upward force to stop the dog is extremely uncomfortable for a dog wearing a head halter. Not to mention, it causes undue strain to your shoulder and arm (odd angle for tension).
You aren't really teaching the dog anything here except where the end of a very short amount of leash is.
Do This.
(yes, I know the dog in the photo is not wearing a head halter, but it was virtually impossible to locate a good example of proper hand position)
Carry your arm as though you were holding an empty suitcase or briefcase. Low and relaxed at your side; yes, even if your dog is a puller---this is where technique comes in to teach the dog what you want.
As with any Tool, unless you are also applying good technique, you aren't really teaching the dog anything. The head-halter is a guiding tool; it is meant to guide the dogs' head where you want it to be. Slapping it on his face and expecting it to work (and it might at first) in the long run, it will fail. The moment you take it off, or it breaks, you and your dog are back to square one: He pulls.
Proper fit is important. This halter is too tight. The dog cannot open its mouth to pant fully.
But he pulls it off his face!
Then you aren't doing your job as a Teacher, or your "trainer" hasn't shown you how to stop your dog from getting free of this device. They SHOULD be relatively easy for the dog to pull off with its paws; if not, you are using it wrong.
This dog is wearing a different style of halter, but you notice how much more comfortable he looks?
His human took the time to adjust him to the device, and it is fit almost perfectly. The strap that lays just behind the ears should probably be a bit more snug, so that it stays behind the ears.
If your "trainer" has told you to tighten this thing so much that the dog cannot slip out of it, you are strangling him with the head strap.
This is a better example of a properly fit head-halter. Notice the dog can open its mouth to its full extent. Your dog should be able to bark, bite, pick up a tennis ball, or chew on a bone in a halter.
Walking in a halter requires technique in order to actually teach the dog something other than "with this on my face, I am incapable of pulling.....you just wait 'til I'm free."
While I applaud this Human for taking an extra step (the double ended leash attached to both a harness and a halter---sometimes it's necessary) this Human also has the harness on backwards, and the dog is obviously STILL leading the walk.
This is a great example of a Human allowing the device to do its job, but not doing the job of Teacher. The moment this dog is out of this contraption, this animal will Still Pull.
If your dog is wearing a head-halter, it is not time for social interaction. He may be around other dogs, but it is time for him to focus and respect you on the walk, not to playfully interact with limited physical responses. You are crippling his natural way of being, and he could get himself into a fight; the head-halter restricts natural body posturing and reactions.
NEVER USE A HEAD HALTER ON A BRACHYCEPHALIC (a dog with a smushy face, or extremely short muzzle) DOG !!!!!!!
These dogs have enough trouble breathing due to their shortened nasal passages. The extra pressure of the head halter's design (and I don't care what your "trainer" said) can cause them some serious issues.
So what's this "Technique" I have to use with this halter thing now?
Well, begin by making sure your dog actually likes wearing it. Don't just go to the store, slap it on and go: You'll get a lot of diving, pawing, rolling on the ground, sometimes they just give up and lay down, etc. Rather, find the right size for your dog, then spend the next few days putting it on, feeding your dog something yummy, or giving them a great toy for a few moments, then take it off. Dinner is a great time to do this, if your dog loves eating. Just put the halter on, feed your dog, and take it off.
Once your dog is used to wearing the halter, and has been rewarded for putting up with you sticking that weird thing on his face, it is time to enroll in a class geared towards your goals.