Because you pay out of pocket up to the deductible with a HDHP, people think twice about going to the doctor for minor stuff that they normally wouldn't think twice about under a copay PPO. When one needs to have a procedure done, they shop around and (likely) take their business to the lowest-cost option. This creates competition between healthcare providers (MD offices, hospitals, etc) which lowers the overall cost of care.
I don't think people should have to think twice about going to a doctor. You never know when that something you think is minor might be something major.