There is no such thing as "free" health care because someone is always going to be paying for it. In this case it is the government that pays for the health care, with our taxes. So end the end we are still paying for it. The difference is that it costs less because fees are more controlled. This is good because doctors make too much anyway!
Under the current private insurance system, 1) patients who cannot afford good medical insurance, even under Obamacare are forced to forgo necessary medical treatment. With single payer insurance, these folks would be able to get the care they need. 2) doctors spend amazing amounts of time justifying their actions to insurance companies, talking with folks with less training and no direct interaction with the patient. Single payer care has a chance of limiting this problem.