Draw reins are a training tool that should be used with a large dose of equestrian tact. Used correctly, they can be extremely helpful in your training process to point you and your horse in the right direction as long as you are mindful of the way they function. Draw reins are only an aid to training your horse. They do not replace the solid training steps that accomplishes a balanced, supple, obedient and happy horse. When overused they can influence your horse to stay behind the contact, overbend the neck, and impede his desire to go forward.
Draw reins run from the girth through the snaffle bit and to your hands. This creates a pulley effect; your hands on the draw reins are now twice as powerful. As a result, they must be used very subtly, and lightly. You combine the draw reins WITH your normal reins, and never use the draw reins by themselves. So you will have two sets of reins in your hands. To avoid using the draw reins too strongly, place the draw rein between your middle and ring finger, and the snaffle rein between the ring finger and pinky finger as usual. Using them this way, the same way you hold the reins of a double bridle, you are less likely to overuse, or pull too much on the draw reins. The weight of the draw reins alone is often enough to get the results you are looking for.
If you try to use the draw reins by themselves, you will end up pulling the horse's nose in towards the chest too much resulting in an overbent horse. And the worst case scenario is you pull the horse's nose so far into the chest, you lose any means to apply the brakes. The western trainers call this, "swallowing the bit." The rider loses the ability to control a horse with the action of the bit if the horse puts his head straight up in the air, but also if the horse puts his head curled into his chest and there is no further place to pull.
However, draw reins can be very beneficial when not overused. Sometimes, it's just the thing that is needed to teach your horse to give to the request of the hands and flex at the pole. It can help to point your horse in the right direction of a round neck and back, and then the real training continues when you take them off. They can also help with some control issues if your horse gets a bit hot when you ask for a flying change for example. Draw reins should not be used outside of the arena or on a trail ride or while jumping. They are too restrictive to jump with, and may get caught on something during a trail ride through the brush.
Draw reins can be a useful training tool for a horse that is resistant to give to the pressure of the bit by avoiding flexing at the poll or jaw. This flexion of the poll or jaw will lead to the horse becoming round in the neck and back, and encourages a more balanced, softer ride. But as soon as the rider can feel the give to the light use of the hand, then it's time to take off the draw reins. Otherwise, the rider becomes dependent on them and cannot get the round, light feeling without them.