“Extraordinary claims require extraordinary evidence” is usually just another way of saying “That seems difficult for me to believe.”
The question is then: why is it difficult to believe?
A few key points are relevant.
1. Much evidence can be re-interpreted, given a theory choice.
For example, much of the evidence that supports a geocentric theory can be re-interpreted to support a heliocentric theory, given a few key, conceptual shifts. A claim may seem extraordinary because those few key, conceptual shifts are not there. Alternatively, the evidential debate will then be thrust back upon the evidence for or against some proposed conceptual shift that allows for a re-interpretation of existing data.
What this means is that the evidential ‘core’ of a theory, when compared to a competing theory, might be much slimmer than supposed.
2. Often, people are simply unfamiliar with certain evidence. The person who proposes a theory may not be familiar with certain evidence, or the person who finds it difficult to believe may not be familiar with certain evidence.
3. Often, people who think a theory is correct might be less critical at evaluating evidence that seems to corroborate it. This can be thought of as a ‘founder’ effect.
For example, is a recent tornado further evidence of global warming? One way to evaluate that is to check the frequency of tornados compared historically. Yet, if one already believes there is global warming, one might be less inclined to rigorously test that hypothesis, and more inclined to rigorously test competing hypotheses.