Let me take a crack at it with a simplified example.
Say you have a layer in a precomp and the precomp is in a main comp. In the precomp you have the layer scaled to 25%. In the main comp you have the precomp scaled to 400%. So, in the main comp your layer will appear to be at 100% scale, but it will look like crap because it has been scaled 400%. Unless… you turn on the continuously rasterize switch for the precomp.
With the switch on, when After Effects displays the layer in the main comp is says “Hmmm, scaled down by 25% then up by 400%. How about if we just leave it at 100%?” (That’s what it says – I swear I’ve heard it). So basically it does the minimum transformation necessary to arrive at the final result, which results in optimum sharpness.
The switch performs a different (but similar) function when used with Illustrator layers. Instead of rasterizing the vectors and then scaling (as usual), AE will first recalculate the scaled vectors, then rasterize, maintaining the layer’s sharpness.
Now go read the Meyers’ book. 🙂
Dan