This pithy saying can actually be applied using Bayes Theorem.
While the idea that a sufficiently outlandish claim requires a lot more compelling evidence is quite intuitive, it can be quantified nicely with probability theory in a Bayesian framework. In short, sufficient evidence must be capable of raising a highly improbable claim to be highly probable - and the more improbable the evidence, the better. By application of Bayes' theorem, it's possible to show this in action mathematically.
Assume, for instance, someone claims to be able to predict what way a coin will land almost perfectly. We know this is an extraordinary claim, so we'll say that just by guessing if the person is telling the truth or not that it's a million-to-one chance. In reality, the number would be even more improbable, but this can be used for illustration. So we ask them to demonstrate the skill. They're
almost perfect, so let's assume they guess right about 90% of the time - this allows them the opportunity for their skill to mess up once in a while, but still prove to be pretty good. This gives us all the information we need to know to actually quantify
how extraordinary the evidence must be.
Consider if they guessed a single coin toss correctly. The odds of guessing by chance is a mere 50%, or 50:50.
A single coin toss doesn't improve our odds very dramatically. The evidence just isn't
extraordinary enough - you can correctly guess a single coin toss correctly 50% of the time with no special skills involved. It all rests on how improbable our evidence, P(B), actually is and a 50:50 chance isn't particularly improbable. For two coin tosses P(B) becomes 0.25, and for 10 coin tosses it comes to roughly 0.00097. Plugging those numbers in Bayes' theorem gives us a probability of genuine skill (given P(A) of a million-to-one) of around 0.0009, which although still small is a considerable improvement on that original million-to-one chance. By 20 or so correctly guessed coin tosses, the skill is starting to look a lot more genuine.