Here's where those increased costs come from:

Increased drug costs and higher Medicare premiums. By repealing the Affordable Care Act, the Romney-Ryan plan would raise health care costs in retirement by $11,000 for the average person who is 65 years old today.
Increased long-term care costs, including increased costs for nursing home care, because of cuts to Medicaid. A substantial share of Medicaid spending pays for health care costs for Medicare beneficiaries. The Romney-Ryan Medicaid cuts mean a loss of over $2,500 annually for seniors currently on Medicare who also rely on Medicaid. Unlike the Medicare voucher system that would begin in 2023 the cuts to Medicaid would begin almost immediately.