It does this by eliminating a very inefficient layer of our current system (insurance companies), and by having there replacement for that (the government) negotiate on drug prices (how much savings there is the reason there are two estimates). Currently insurance companies almost have a negative incentive to push down drug prices (their profits are limited to a percentage of total spending, and most large companies are pushing against that limit).
Most of the cost control pressures in our current system come out of Medicare/Medicade, and this would widen that out to the whole system. That in turn would wedge open the door to pushing on the other drivers of the cost spirals: hospital administration, new expensive drugs that are not worth the additional costs, doctor salaries ballooning, and the broken system between malpractice insurance and dysfunctional enforcement against malpractice.