The documents were scheduled to be released in 2017 unless the President stopped it. The 2017 date (specifically October 26, 2017) came from the JFK Assassination Records Collection Act. This act was passed by Congress and President Bush signed it on October 26, 1992.
All this information and more can be found on the JFK Assassination Records Processing Project page.
October 26th, 2017 passed, and some of the records were released. But not all of them... As I said, only the President could prevent documents from being released, and that he did—at least for now. Here's what Trump said in a memo on October 26th:
The American public expects — and deserves — its Government to provide as much access as possible to the President John F. Kennedy Assassination Records (records) so that the people may finally be fully informed about all aspects of this pivotal event. Therefore, I am ordering today that the veil finally be lifted. At the same time, executive departments and agencies (agencies) have proposed to me that certain information should continue to be redacted because of national security, law enforcement, and foreign affairs concerns. I have no choice — today — but to accept those redactions rather than allow potentially irreversible harm to our Nation’s security. To further address these concerns, I am also ordering agencies to re-review each and every one of those redactions over the next 180 days. At the end of that period, I will order the public disclosure of any information that the agencies cannot demonstrate meets the statutory standard for continued postponement of disclosure under section 5(g) (2)(D) of the President John F. Kennedy Assassination Records Collection Act of 1992 (44 U.S.C. 2107 note) (the “Act”).
The NY Times: J.F.K. Assassination Files Released, Highlighting Hoover, L.B.J. Among Others
180 days later, on April 26, 2018, some more documents were released.
As of now (2019), here's what's happening: "The President [Trump] has determined that all information that remains withheld under section 5 must be reviewed again before October 26, 2021 to determine whether continued withholding from disclosure is necessary."
Obviously, this wasn't around in 1968, when the interview was given. I'm not sure where the 2038/2039 date comes from exactly, but Wikipedia says this:
The unpublished portion of those records was initially sealed for 75 years (to 2039) under a general National Archives policy that applied to all federal investigations by the executive branch of government, a period "intended to serve as protection for innocent persons who could otherwise be damaged because of their relationship with participants in the case.” The 75-year rule no longer exists, supplanted by the Freedom of Information Act of 1966 and the JFK Records Act of 1992.
Wikipedia: Warren Commission
According to the Final Report of the Assassination Records Review Board, in 1991 the release date was going to be 2029:
While the movie [JFK, 1991] was largely fictional, the information that Stone conveyed in the movie’s closing trailer was true: the HSCA [United States House Select Committee on Assassinations] had reinvestigated the murder and issued a provocative report, but their records were sealed until the year 2029.