My question would be why rather than how. After working in the industry all my life I've never been able to figure out how anyone can possibly run an airline at a profit. They obviously do or nobody would waste their time trying.
It also depends what you mean by airline. Virgin Atlantic started life as British Atlantic Airways in 1984. It started in a small way, its first regular transatlantic service (Gatwick/Newark) with a single aircraft in June 1984 gradually expanding into the international airline it is today. Richard Branson sold off his other interests including his Virgin Records empire to help finance it.
http://www.virgin-atlantic.com/en/gb/allaboutus/ourstory/history.jspMany airlines of all sizes have sprung up since then all over the world. Some have fallen by the wayside while others have succeeded. One example is easyJet, which was again started (in 1995) by a wealthy entrepreneur as a small operation with only one or two aircraft. It was originally run from a small office at Luton Airport.
http://www.easyjet.com/EN/About/Information/infopack_keyevents.htmlI've lost touch with it all now & don't know the present state of the industry or which new airlines have appeared over the last few years.