I watched this video with great interest, but there were a lot of annoyances in it and misleading statements:
First let's get the annoyances out of the way. The video shows how the studio was setup for Mechner's talk, but the main background computer is an IBM PC. Mechner is talking almost exclusively about development of Prince of Persia for the Apple II, which was the original development machine for the game. They have the wrong machine in the background! Moreover, Prince of Persia is almost unplayable on an IBM PC, despite being almost invariably more advanced than even an Apple IIe Enhanced.
Additionally, at several instances in the video, footage from the Apple II version is shown but Adlib music from the PC version is played. The Apple II version was limited to clicking the internal speaker, it did not support sound cards but you cannot expect a casual viewer to know that.
Now those flaws are solely on the producers of the video, but let's turn to Mechner's comments.
"At that time the Apple was the #1 game plaform" - Debatable in the early 1980s when Karateka was first being conceived, but defensible if you add the word "computer" before "game". By 1985 when Mechner first started developing Prince of Persia, the Commodore 64 would have been a stronger contender for the #1 computer gaming platform.
"[The Apple II] only had four colors" - This statement is true only if you limit color to exclude black and white and ignore the 16-color capable Graphics and Double High Resolution Graphics modes. Mechner used purple and green very seldomly when Prince of Persia's used High Resolution Graphics mode, which is the only mode in which gameplay occurs. He did use Double High Resolution Graphics mode for the opening title, credits and title captions (5 screens).
"Everything had to fit into 48K of memory" - Only on an Apple II or II+ without a Language Card. Karateka ran in 48KiB Apple II and II+s, but Prince of Persia required 128KiB of RAM. The video make not make it clear that Prince of Persia loads levels from a floppy disks, copying the data from the disk to RAM, so the whole game did not have to fit in 48KiB but 140KiB x number of disk sides used. Even Karateka loads only a portion of itself at any given time. Prince of Persia used both sides of a floppy disk.
"I found a way to take 12K of memory that was hiding in the Auxillary Memory Card of the Apple II" - The idea that some memory was hiding from a programmer so familiar with the Apple II platform is ludicrous. The Apple II, even the IIe and the //c were very well-documented machines. It is true that they had a fairly Byzantine approach to memory mapping (required for backwards compatibility), but the intricacies of the bankswitching should have been well-known before 1989. Prince of Persia requires a 128KiB machine and not just because it supports Double High Resolution Graphics. Even though the gameplay was limited to High Resolution Graphics, the game ended up using almost every byte of memory available to an Apple IIe with 128KiB of RAM according to Mechner's Prince of Persia Technical Information document : http://www.jordanmechner.com/downloads/popsource.pdf
What I think Mechner meant that the 12KiB was "hidden" was not that he did not know about it, 12KiB being way too large to hide, but hidden in the sense that it was tricky to use and therefore unavailable without a great deal of effort. The area I believe he is referring to is the 12KiB provided by Language Card functionality on the Auxillary Memory Card, which requires bankswitching on top of bankswitching to access.