Online Etymology Dictionary says:

unquote (v.)

1915, in telegraphs, where punctuation had to be spelled out and quote and unquote were used in place of the quotation marks; from un- "reverse, opposite of" + quote. Quote unquote together to indicate quotation of the word or phrase to follow (often with ironic intent) is from 1942.

To the question of the construction/placement, this linguistics professor describes quote/unquote as a "compound prefixal particle", and he explains its usage as a concern by the speaker for improved effect on the listener:

…if you are doing scare quotes in speech, and thereby taking your chances with the listener's short-term memory, chances are you're targeting only one word or phrase—one phonological unit in any case—and you don't really need to mark the end; it's obvious. All you really need is a marker at the beginning to warn your parter to listen ironically.

To the question of origin, Online Etymology Dictionary says:

unquote (v.)

1915, in telegraphs, where punctuation had to be spelled out and quote and unquote were used in place of the quotation marks; from un- "reverse, opposite of" + quote. Quote/unquote together to indicate quotation of the word or phrase to follow (often with ironic intent) is from 1942.