Teaching your horse to willingly put the bit in their own mouth is easily done using positive reinforcement and shaping.
Positive reinforcement is something added to the horse's environment that makes a behaviour more likely to happen again in the future. In horse training, it is usually a small tidbit of tasty food, given immediately after a desired behaviour occurs. Shaping means to take a complex end goal behaviour, like having a horse put a bit in their own mouth, and breaking it down into smaller behaviours - or 'successive approximations' - that build towards that end goal. As you can see in this clip, when combined, the horse receives positive reinforcement every time he successfully completes a successive approximation towards the end goal. Effective trainers usually make a 'shaping plan', or a list of the behaviours that will build towards the end goal.
Once the horse is reliably performing a successive approximation, it is changed slightly, and reinforcement is given when the horse successfully completes the new successive approximation. For example, when this horse Calcite was reliably sticking his nose into the open bridle, the next successive approximation I aimed for him to do was to move his muzzle closer to the bit. When he did that, I indicated that’s exactly what I wanted by making a clucking nose and immediately giving him a treat. (One of the first lessons taught in this type of training is that a tongue cluck or other specific sound marker means ‘Yes! A treat is coming!’)
Just like us, horses learn from the consequences that follow a behaviour. If the consequences are desirable (like getting a treat) the behaviour is likely to be repeated in the future. This type of training says very clearly ‘Yes, that’s exactly what I wanted you to do!’ to the horse, and it eliminates the need for us to use punishment in training.
All animals, humans and horses included, would rather be told ‘Yes!’ and receive a desirable consequence when learning how to perform a new behaviour. It's important to remember that the horse doesn't know what our goal is during training. They are essentially guessing when they try out a new behaviour, to see if it is the one that will result in a desirable consequence. Being told 'No!' and receiving an undesirable consequence for ‘wrong’ behaviours when trying to learn a new behaviour is stressful. This stress usually results in the animal giving up trying to find out what behaviour results in a desirable consequence, for fear of receiving the undesirable consequence instead. This type of training can also damage the trust between trainer and animal learner, and should be avoided.