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.
" Hypertext is simply a non-linear way of presenting information. ... This is accomplished by creating "links" between information. These links are provided so that readers may "jump" to further information about a specific topic being discussed (which may have more links, leading each reader off into a different direction). "
The given quote "maybe the two different worlds we lived in were not so different. We saw the same sunset" can be found on Page 40 in the book "The Outsiders" by S.E. Hinton.
the page number may differ considering the publication and version one is using.