The On-Access word tells everything
When you access some file,avast! checks it. Even when you browse webpages,some files are stored into browser cache and avast! checks them.
To hide message you're refering to you need to do the following:Left click on "a" ball next to the clock,select Standard Shield provider and click Customize... button on the right (if you don't see Standard Shield then click More detail... button below).
Now click on Advanced tab (last one) and uncheck(remove checkmark):
"Show detail info on performed action"
This should do the trick