Based on what House Budget Chairman Paul Ryan has said, the final plan (to be released tomorrow) will call for transforming Medicare from a government-run insurance plan into a system of competing private plans, from which beneficiaries will choose.
But simply switching from public to private insurance doesn't actually make Medicare more efficient. If anything, the very opposite is true. Medicare has lower administrative overhead and so, benefit for benefit and person for person, it's actually cheaper than private insurance--which means that privatizing Medicare should increase costs, all else being equal.
So how would the Republican budget save money? Via Krugman:
...those savings will come entirely from limiting the vouchers to below the rate of rise in health care costs; in effect, they will come from denying medical care to those who can’t afford to top up their premiums.
Republicans do have a more serious and (somewhat) more legitimate argument for why a plan like Ryan's would save money better than traditional Medicare. But that's a topic for another post.