Their arguments are of the form "this statement could be false."
If you evaluate it as a true statement, you have no problems. It could be false; you have to evaluate it for yourself instead of trusting some authority.
It's only if you assert that it's certainly false that you have a problem. Because then it's clearly true -- since otherwise these authorities would be telling you something false, which proves their assertion true.
Put another way, it can get you from the undesirable position of blindly trusting authorities to the desirable position of questioning them, but not the other way around. Which is the intended result.