He said they drew a lot of inspiration from Woz on the hardware side. Woz was well known for employing lots of little hacks to make things more efficient, and the Macintosh team had to apply the same approach to software.
My guess is it asks each application that overlapped those areas to redraw only those areas (in case the app is able to be smart about redrawing incrementally), and also clips the following redraw so that any draw operations issued by the app can be "culled". If an app isn't smart and just redraws everything, the clipping can still eliminate a lot of the draw calls.
The very characteristic horizontally stretched graphics of the Atari 2600 are due to this - the CPU was actually too slow, in a sense, for the electron beam which means your horizontal graphic elements had a fairly large minimum width - you couldn't change the output fast enough.
I strongly recommend:
https://en.wikipedia.org/wiki/Racing_the_Beam
... which goes into great detail on this topic and is one of my favorite books.
Per-pixel sorting while racing the beam is tricky, game consoles usually did it by limiting the number of objects (sprites) per-line, and fetching+caching them before the line is reached.
There was a discussion here a couple of weeks ago (with a typo in the title): https://news.ycombinator.com/item?id=44110219
The Smalltalk-76 MVC user interface that the Apple people saw only ever updated the topmost window which, by definition, was not clipped by any other window. If you brought some other window to the front it would only then be updated. But since nothing ran in the background it was easy to get the wrong impression that the partially visible windows were being handled.
Bill's solution had two parts: one was regions, as several other people have explained. It allowed drawing to a background window even while clipping to any overlapping windows that are closer. But the second was PICTs, where applications did not directly draw to their windows but instead created a structure (could be a file) with a list of drawing commands which was then passed to the operating system for the actual drawing. You could do something like "open PICT, fill background with grey pattern, draw white oval, draw black rectangle, close PICT". Now if the window was moved the OS could recalculate all the regions of the new configuration and re-execute all the PICTs to update any newly exposed areas. If the application chose to instead draw its own pixels (a game, for example) then the OS would insert a warning into the app's event queue that it should fix its window contents.
In parallel with Bill's work (perhaps a little before it) we had Rob Pike's Blit terminal (commercially released in 1982) which added windows to Unix machines. It had the equivalent of regions (less compact, however) but used a per window buffer so the terminal would have where to copy newly exposed pixels from.