It is impossible, and unnecessary, to know the motivation for using "
in element content, but possible motives include: misunderstanding of HTML rules; use of software that generates such code (probably because its author thought it was “safer”); and misunderstanding of the meaning of "
: many people seem to think it produces “smart quotes” (they apparently never looked at the actual results).
Anyway, there is never any need to use "
in element content in HTML (XHTML or any other HTML version). There is nothing in any HTML specification that would assign any special meaning to the plain character ” there.
As the question says, it has its role in attribute values, but even in them, it is mostly simpler to just use single quotes as delimiters if the value contains a double quote, e.g. alt='Greeting: "Hello, World!"'
or, if you are allowed to correct errors in natural language texts, to use proper quotation marks, e.g. alt="Greeting: “Hello, World!”"