Since graduating from college in I believe 1935, Richard Nixon had been a winner his entire professional career up until 1960 when he lost that presidential election to John Kennedy. So he was going through a very difficult stage in his life that lasted from losing that election until probably as late as 1966 when he had a lost of success in 1966 campaigning for Congressional Republicans. Even though by 1963 he went back into the private sector as a New York lawyer and public speaker. But politics is what made him very happy.
Because of all these things and the mounting frustration that had built up of not only being out of public office for the first time since 1946 when he left the Vice Presidency in early 1961 and losing his second major election in 1962 in just two-years, Nixon just sort of went off and took his frustrations out on the press and let them have it. Calling this press conference his last, which of course it wasn't. I just don't think he put a lot of thought into what he was going to say here and went off instead.