What does it mean when it says under a journal article that it has been communicated by "XYZ" where XYZ is not the author but some other scholar with a very strong reputation? What is the relationship to the actual author and/or the content? Is this some sort of seal of approval to get results out and known quickly? (I am specifically wondering in the context of mathematics and mathematical physics.)
Generally, XYZ refers to the editor that handled the paper at the journal.
See this question on Mathematics.SE and its very good answers for full details, which I will summarize below. It should be noted that this information is part of the journal format, and added by the publisher itself (along with the publication timeline).
Some journals published by learned societies or national academies require that “communications” be presented (or sponsored) by a member of the society. This was the case, for example, of the PNAS (Proceedings of the National Academy of Sciences) until July 2010; the top of an article looked like this:
Some journals use this formulation to denote the handling editor: the one who makes the editorial decision (or recommendation to the full editorial board), after having selected referees and received the referees' reports.
This is not common practice: most journals do not indicate who the handling editor was for a given article.
It is important to note that "Communicated by" can mean a direct submission of a paper by a scientist who is not (in most cases) directly involved in the paper itself. It was designed as a way for established scientists to give a "leg up" to their younger colleagues by allowing them to circumvent the normal review process. This means you may have to give these types of papers a bit more scrutiny as a reader.
PNAS is the publication where I have most commonly seen "Communicated by" publications, but this feature was phased out in 2010:
Until July 1, 2010, members were allowed to communicate up to 2 papers from non-members to PNAS every year. The review process for these papers was anonymous in that the identities of the referees were not revealed to the authors. Referees were selected by the NAS member. PNAS eliminated communicated submissions through NAS members as of July 1, 2010, while continuing to make the final decision on all PNAS papers. (wikipedia)