Why are horses broken in when they are grown up? Wouldn't it be easier to do it when they are little?
Horse shouldn't be ridden or in heavy training when they're less than two years, as it makes them more susceptible to injuries and musculoskeletal conditions. The body is still growing and developing and doesn't need the extra weight or stress. Much like over exercising a puppy. Interestingly horses that are not worked (ridden, trained for domestic use) until they are older, than two are more susceptible to injury. Perhaps the body adjusts to the type of work and stretching as a two year old, while still being supple, but well developed enough to take the impact on joints. (“Young Horses in Training and Injury Risks,” 2001)
The key to breaking in horses is handling and human relationships. Regular handling from birth, building trust is essential for the development of a horse that is going to be worked. Getting used to wearing halters and a lead rope are easier done on a young horse, with a confident bond with it's handler.
Walking foals onto floats, picking up their feet, grooming, worming, checking teeth, these are all things that need to be practiced to get the foal used to it. A calm experienced handler will instil security in a young horse that will take it through it's life and making the breaking in process easier.
These days more people are turning to positive training methods, including breaking in and many people have such strong relationships with their horses, they're able to gradually put weight on their backs and when they're used to that, slip onto the horse. The use of force and fear, and things like whips, spurs, bits are becoming increasingly unpopular.
A young horse that has been mistreated or not handled can be nervous around people and will more likely be more difficult to break in and handle generally, those early months and years are vital. That being said, horses do have a variety of temperaments, so there is no fool proof plan, just methods to optimise the outcome.
Geor, R BVSc, PhD, Dipl. ACVIM (2001)Young Horses in Training and Injury Risks
Adding to Yvettes wonderful answer - the bones of a horse grow until the horse is 6 years old. Expecially parts of the spine (where the rider would sit) need time to grow fully and healthily.
Common handling like hoof care, vet visits, trailer loading, leading, experiencing the outside world should be taught to the horse to make it safe to work with in daily handling.
Light exercising, getting used to a pad or saddle or even a small and balanced rider should - in most professional opinions - be started at age 3.
Riding and hard work however should take the emotional, mental and physical maturity into consideration. This is also very individual, depending on the horse, the handler and the training method.
The slower you start training, the longer your horse will stay healthy as the sources mentioned by Yvonne and me will confirm.