What @waetherman basically said.
You don’t have to be particularly smart to do what Conway does: you just have to be trained to not answer a question you don’t like or can’t/won’t respond to. And have the temerity to keep going, essentially ignoring what the journalist says/asks, to get your preferred message across.
A good way to achieve the non-answer is to say something like: “I can’t answer that. What I can say is…” and then hope the journalist focuses on the answer, rather than what you didn’t say/respond to. Or you can just. Keep. Talking. Whatever nonsense falls out of your mouth, keep going. Pause as little as possible. Or, as a last resort, end the interview. Conway seems to be especially good at the ‘just keep talking with whatever nonsense’ thing.
Of course, in an ideal situation (for the interviewer), you don’t get distracted or misled. You keep hammering away. Ask the same question 20 times, 50 times if you have to. But that’s where you’re willing to risk pissing off the subject, even to the extent that they refuse to speak to you and/or the organisation again (possibly indefinitely) or, as said earlier, end the interview. But that opens up a whole other line of possible questions - why so sensitive, what is being avoided, etc, etc.
Disclaimer: (Australian) journalist/media professional for 26 years. With media training when I was in a corporate role.