The answer to this question clearly explains the standard rule that when you have multiple quoted paragraphs, each new paragraph starts with an opening quotation mark, but only the final quoted paragraph has a closing quotation mark at its end.
This Wikipedia article on Quotation Marks agrees:
Quotation marks are used for multiple-paragraph quotations in some cases, especially in narratives. The convention in English is to give opening quotation marks to the first and each subsequent paragraph, using closing quotation marks only for the final paragraph of the quotation [ . . . ]
However, neither explains why this is the standard practice. What good does it do? What is it trying to avoid? What harm would occur if it were ignored and people put both opening and closing quotation marks on each adjacent quoted paragraph?
Best Answer
“That seems like an odd way to use punctuation,” Tom said. “What harm would there be in using quotation marks at the end of every paragraph?”
“Oh, that’s not all that complicated,” J.R. answered. “If you closed quotes at the end of every paragraph, then you would need to reidentify the speaker with every subsequent paragraph.
“Say a narrative was describing two or three people engaged in a lengthy conversation. If you closed the quotation marks in the previous paragraph, then a reader wouldn’t be able to easily tell if the previous speaker was extending his point, or if someone else in the room had picked up the conversation. By leaving the previous paragraph’s quote unclosed, the reader knows that the previous speaker is still the one talking.”
“Oh, that makes sense. Thanks!”