It seems to me that when a statement like "claims demand evidence" is used, it is meant to imply that a particular claim needs evidentiary support if it is to be accepted. Vorlon already pointed out that in our day-to-day lives we accept a lot of claims without evidence, but that certain claims would be regarded as unlikely or dismissed outright without sufficient reason to believe. It doesn't mean that those mundane claims are true, but claims that appear sensible are going to face fewer challenges.
"Well, evolution is a theory. It is also a fact. And facts and theories are different things, not rungs in a hierarchy of increasing certainty. Facts are the world's data. Theories are structures of ideas that explain and interpret facts. Facts don't go away when scientists debate rival theories to explain them. Einstein's theory of gravitation replaced Newton's in this century, but apples didn't suspend themselves in midair, pending the outcome. And humans evolved from ape- like ancestors whether they did so by Darwin's proposed mechanism or by some other yet to be discovered."
-Stephen Jay Gould
-Stephen Jay Gould
