A detail which provides specific evidence to <em>develop an author's claim</em> is the proof which backs up the claim of a person and validates their claim by making use of either:
Statistical data
Figures
Scientific reports, etc
A supporting detail or evidence is important as it helps to provide evidence which shows that a claim is valid.
For example, if a person states that America has more blacks than Hispanics, he would have to bring statistical data from a trusted source to prove his claim.
Please note that your question is incomplete so I gave you a general overview to get a better understanding of the concept.
