She doesn't. She can't actually disprove or rebut any of the mounds of evidence that she is and was the most ignorant and unqualified person ever put on a national ticket. All she does is deny, deny, deny:
“I had been warned not to watch it,” said Ms. Palin of the “60 Minutes” segment in question. That segment dealt with a new book on the campaign that alleges, among other things, that Palin did not know why North and South Korea are two different countries.
“That is a lie,” said Palin.
Host Bill O’Reilly of “The O’Reilly Factor” said that Palin couldn’t have bested Vice President Biden in a debate if she were really that dumb, and offered her his show as a base for future anti-“60 Minutes” offensives. “You now have a forum here at Fox News where you can immediately neutralize ’60 Minutes’,” Mr. O’Reilly said.
O'Reilly effectively backs Palin's claim on the basis that her crammed, force-fed burbling of talking points in the Biden debate somehow refutes the idea that she hasn't the slightest clue what goes on in the world or the slightest knowledge of history outside of sports. And he is essentially pledging that News Corporation will advance her lies and spin, as it did by publishing her book, and protect her from any real scrutiny that a political candidate deserves.
Since Fox is a propaganda operation and not a journalistic enterprise (Shep Smith excluded), they don't push back. What FNC is giving her is a platform to lie unchecked by any journalistic ethics.
I do not believe that this means she is out of politics. Au contraire. FNC and the RNC are effectively the same operation (and Harper Collins, which published her fiction as non-fiction with no fact-checking or editing is also part of NewsCorp). Her new job is running for office via the chief propaganda network for the red states. The strategy is obviously to focus entirely on the base, demonizing the president and anything he does, exploit economic malaise, and then get back to power on a wave of frightened white ressentiment with Palin as the hood ornament one more time.