There does not seem to be any correlation between the animation of the tray icon, and Avast activity.
If I hover over it . . . / If I click to it and go to Home . . .
. . . it doesn’t say what it’s doing, you just see “Everything is good.”
Are you supposed to look somewhere else for this info?
There is indeed a corelation between the animated icon and avast activity. It tells you that avast is currently scanning files/links in the background. Of course to know this, you have to check the statistics window under ‘Realtime Data’
Its one of the aesthetic changes (earlier it was a revolving 3D icon) avast made that was pleasant to the eyes. 
The icon is only showing animation if avast is scanning something.
Your assumption that there is no relation between the animating icon and avast activity is totally wrong.
What it is scanning/doing?
It can be a file from a application you are starting, it can be internet traffic etc etc.
If you want to see what was scanned/analyzed check the log files.
Got it, thanks. In Win 8.1 it looks like they’re under:
C: \ ProgramData \ AVAST Software \ Avast \ log
Would be nice if this could be presented in some user friendly, informative way, like some statistics, and easily found.
Open avast! > Statistics. You can see what files and sites avast! has checked lately. Not many are shown, but you can get an idea.