Effect of pharmaceutical Interferons on the immune system

If one can carry a virus for many years and the body doesn’t recognise it, how come when we administer synthetic interferons, which the body already has, the virus can be eliminated?

What puzzles me is: how does adding extra interferon improve the recognition abilities of the immune system towards that virus, when it has been languishing for decades in its host, untouched by the effects of the signalling of the host's own natural interferons on his/her immune system? What does it do to the, I’m guessing here, ‘killer’ cells that gives them that ability to recognise the virus?

I’m thinking of the hep c virus in particular since interferon-alfa works quite well against it.