It's from the audiobook version of the book "The Nitpicker's Guide For Classic Trekkers." The audio version was released in three separate parts consisting of three audiocassettes each (I think). Koenig, Nichols and Takei were narrators.
Edit to add: to clarify, the video was put together by the Youtuber who posted it, using selected snippets from Nichols' "Nitpicker's Guide" reading as narration for the video.
Kor