I'm not really in agreement with some of the other responses so perhaps this is an informal or slang English that can be interpreted multiple ways. That said, I would interpret:
“The best cure—quote, unquote—for aging is slowing disease,” Daniel Kraft, the chair of [...]
to mean that the word "cure" is being used in a way that is perhaps not technically correct. In this example the reason for that is that there is no "cure" to aging. Not only is aging not really a disease which could have a cure, even if it were there is not anything that does actually cure aging.
There are multiple reasons for using quotations. One use is for referencing a word or sentence that was written or said before, like I just did in the above paragraph.
Another common usage is to convey uncertainty, skepticism, or sarcasm about the quoted material. This has even become part of spoken English, where people will make "air quotes" using their fingers to convey the same kind of meaning used in writing. Hesitant as I am to cite to Urban Dictionary, it seems relevant here. Air quotes.
Using quotes for skepticism and ambiguity is similar, but the context indicates that in your example it was probably more uncertainty. That's because it appears that the speaker was making an argument and it would be counterproductive to express skepticism about something you are simultaneously arguing for. If the context seemed critical it might be sarcasm. In short, you can't understand the usage of quotes in this type of circumstance unless you can understand the context.
And how did it come to be? I don't know, probably by an evolutionary or iterative path like any slang. There are other phrases that it could have evolved from, such as "And I quote, ..." or "...open quote, 'I despise grammar,' end quote." In particular I suspect the latter, which is fairly wordy and clunky, simply got abbreviated to "quote, 'I despise grammar,' unquote." Then it later took on the other meanings I described above and became an adjective rather than oral punctuation.
The OED gives some examples that might back this up:
Used in actual and reported speech to represent the beginning of a passage that one is quoting or purporting to quote; freq. in quote..unquote (also quote-unquote, quote, unquote, etc.) (representing opening and closing quotation marks around the quoted word or phrase).
1918 Bridgeport (Connecticut) Telegram 11 Dec. 2/3 Title of picture to be quote Watchful Waiting unquote.
1921 Chicago Tribune 6 July 19, I knew her when she was a quote bear unquote period.
Then by 1992 we see:
1992 New Scientist 19 Sept. 14/2 The most serious land reclamation problems in Wales are lands which have already been, quote, reclaimed.