One often reads about scientific (or other) claims that are made, which can be disputed. For example, the claim that applying cucumber peels fermented in mayonnaise to one's joints nightly for three months cures rickets or what-have-you. Can утверждение be translated as 'claim' in these cases or would another word be used? I'm guessing утверждение could also be translated as 'assertion'.
You can think of "ли" as of a particle expressing doubt. Thus,
• Это утверждение верно. — This claim is correct.
• Верно ли это утверждение? — Is this claim correct?
• Это ли утверждение верно? — Is this claim correct (not another claim)?
By putting "ли" after "верно", we question "верно". By putting "ли" after "это", we question "это".
Why the inversion? I think it has to do with Wackernagel’s law according to which clitics (words depending from other words phonologically — such as "ли" here) in ancient Indo-European languages had to be in the second position of a clause. This rule is almost gone in modern Russian, but placing "ли" on a position other than second sounds overly colloquial for DL, although it’s not ungrammatical anymore.