Riding a horse without reins is not as difficult as you might think, and not a skill only achievable by horse training guru's. Riding a horse without reins does take skill and ability to ride with your seat and legs and not rely on your hands, but it's not an impossible task for the average rider. The correct way to ride a horse is off the seat and legs and not the reins, so if you riding your horse correctly in the first place, it should not be too difficult to ride without reins.
To train your horse to go riding without reins you first need to start working in smaller increments. For example, before just removing your reins and riding out into an open field, you should begin working rein-less in a small area such as around pen or small arena. Instead of removing your reins completely immediately, simply drop your reins and allow them to lay over the horses neck. Practice turning, stopping, starting, and eventually backing up with no reins. At first you will probably need to eggagerate your seat cues- try using your breath to signal downward and upward transitions. This will require lots of nonverbal communication with your horse. (Of course you can train your horse to ride without reins by using voice cues, but it's much cooler if you don't!) After you've mastered these movements at a walk, try the trot and eventually canter. As you and your horse are communicating better and learning to work without reins, try looping a lead rope around the horses neck and letting it rest there as you work. This allows you to retain a small bit of physical control if you need it. Remain in the small enclosed area until you absolutely trust your horse bridleless. Then you can begin removing your reins completely. Always have someone nearby in case you need help. Riding a horse without reins is something that most intermediate riders can aspire to. Precision and correct movement without reins can be difficult but not impossible. Everyone has to start somewhere! Good luck riding your horse without reins.