I think he's referring to the browser implementation's tendency to cache the look-up of elements with a specific class. That is they keep a list of all the elements for each class. You'll often see comments about caching selectors for performance reason for long selectors. (Usually just telling the programmer not to query for the same thing multiple times in JQuery if it's expensive). It's really not something one has to think about in regards to styling though. When you insert a new node into the DOM the look-ups are fairly quick.
I'd imagine, if it's true that CSS selectors aren't memoized past the first class, that it's done for memory reasons. Storing lists of all the elements that match specific CSS rules and keeping track of them when elements are added and removed could potentially eat up a lot of memory and needless CPU on small devices. (Potentially for very little gain if the queries are infrequent).
Also every browser can implement things anyway it wants under the hood. MDN has this article: https://developer.mozilla.org/en-US/docs/Web/Guide/CSS/Writing_efficient_CSS on some best practices. It's possible other browsers have their own guides.