Idiomatically, "owning a room" means commanding the respect and interest of everyone in it. It is usually applied to situations where a performer or (as here) an interviewee is sufficiently entertaining, thoughtful, and intriguing to make everyone present want to pay close attention to everything he or she says or does.
The expression seems to have gained currency fairly recently, although glimmerings of it are evident from somewhat farther back. One early match for the phrase that hints at its subsequent figurative use appears in Maud Goodwin, Four Roads to Paradise, serialized in The Century Magazine (May 1904):
Before he could shake himself free, Yates was literally cornered by the plain young lady in yellow, and could console himself only by the excellent view of Anne which his position gave him. "She is not really handsome,” he declared; “it is only the way she carries her head and her general air of owning the room and the company."
Yates was right. Anne's manner was labeled hors concours, like the pictures in the exhibitions, and signified that it was his own merits rather than hers which were being decided by her neighbor's estimate.
Similarly, from Stephen Graham, "An Ikon in Your Room," in The Treasury (June 1908):
Certainly the ikon is a power, it gives an atmosphere to its room. It owns the room, or rather it is a Presence in the room. It reminds, it restrains. Outside are the sun and moon and stars, the beautiful creation to remind ; inside the ikon takes their place.
But in a Google Books search, the earliest match for the phrase used in the sense that the OP asks about is from Writers Forum, volume 6 (1979) [combined snippets]:
...collar, but the face was the same. Immediately people were on their feet and moving to the door, reaching for him and laughing, hands pulling at his arms in a tumult of adulation.
He owned the room. The calm easiness with which he viewed crowd showed he knew it. The big sweet smile of the boy next door, the effortless way he moved through the people, shaking hands, calling the names of men who reached over shoulders to slap his back. He had always known he had magic in his face, a gift he could not explain but one he had happily used all his life. And in the bright smile practised affection I could tell he had not lost his touch.