I used them to implement an arbitrary precision math library for an embedded processor that needed to do some precise calculation in an autonomous environment. The embedded processor had no floating point capabilities.

The book showed me the method of how to do it efficiently in a manner that could be adapted to the 16 bit architecture I was working with. It also showed me how to detect and handle overflow situations, as well as when they would happen. It kept me from creating a buggy, error-prone pile of crap code that I'm certain I would have created at the time.

At the same time, it took me a couple of days of study to understand the two or three pages I was interested in. The material was the most condensed material I have consumed in computer science. Like you say, you have to learn a new machine architecture and assembly language to understand the examples. Things you learn from those books come at a high cost in terms of time and effort.

It reminds me of studying the bible, where cross checking, reading different translations, studying hebrew and greek, and the culture of the day are all necessary if you want to get the best understanding you can. The TAOCP books are scholarly articles and almost a kind of shorthand notation of computer science concepts.