"Extraordinary claims require extraordinary evidence." Carl Sagan was not the first who said this, but he was the one who made this principle famous. This is a sensible principle, and it seems that most people follow it most of the time. If, for example, you told someone that you had spent a night at Hotel Palomar San Francisco, she would just believe you. But if you told her that you had spent a night in the White House's Lincoln Bedroom, she would not believe you until you showed her some evidence (probably evidence stronger than just a few pictures, which could easily be fakes).
David Hume seems to be referring to the same principle when he remarks that "we readily reject any fact which is unusual and incredible in an ordinary degree". What he says next, however, is truly insightful:
Yet in advancing farther, the mind observes not always the same rule; but when anything is affirmed utterly absurd and miraculous, it rather the more readily admits of such a fact, upon account of that very circumstance, which ought to destroy all its authority. The passion of surprise and wonder, arising from miracles, being an agreeable emotion, gives a sensible tendency towards the belief of those events, from which it is derived. And this goes so far, that even those who cannot enjoy this pleasure immediately, nor can believe those miraculous events, of which they are informed, yet love to partake of the satisfaction at second-hand or by rebound, and place a pride and delight in exciting the admiration of others. ("Of Miracles")
To put Hume's insight in the language of Sagan's principle, many people act as if extra-extraordinary claims require no evidence at all. Let me extend the above example. Yes, if you told someone that you had spent a night in the White House's Lincoln Bedroom, she would not believe you until you showed her some strong evidence. But if you told her that you had spent a night with an angel at the top of a mountain, who demonstrated his power to you to prove that he's really an angel and then revealed some parts of your future to you, she might just believe you (and she might not even have the same religion as yours)!