It is quite authentic and I believe it's still on display in the National Archives.

To why it was addressed to Kissinger, that one's easy. 3 USC §20 requires it:

The only evidence of a refusal to accept, or of a resignation of the office of President or Vice President, shall be an instrument in writing, declaring the same, and subscribed by the person refusing to accept or resigning, as the case may be, and delivered into the office of the Secretary of State.

I believe that was one of the components of the Presidential Succession Act of 1947, although the Secretary of State has had some very oddball authority provided to the office over the years; while in office, John Quincy Adams woke up at 4 am for a year to work on a report to Congress about Weights and Measures - that it promptly buried when he finished!