My understanding is that 2 things happen.
The outer lipid containing layer is attracted to the water,and therefore breaks up in the presence of water and detergent, leaving the virus without any protection - this is a massive over simplification, and completely wrong, as most people tend to be multicellular life forms (i.e. mostly having more than one mobile phone), and viruses are possibly not even life forms let alone multicellular, but think of how nonviable you would be, and how difficult it could be to work, or even reproduce, if all your cells lost their cell walls.
But the principle, and effect, according to my understanding, are similar.
Like dipping an egg in strong acid, causing the shell to breakup/dissolve - any resulting chickens aren't going to be.
Water both makes the break up of the lipid layer possible, and if moving washes away the resulting chaos.
My virology fu is somewhat weak but my understanding is that the 'keys' that the virus needs to enter cells are contained in/on the lipid layer, so even if the virus was to survive the removal it would be unable to enter any cells.
I, personally, don't think the rubbing will make that much difference, it serves to ensure distribution of the detergent, and it's probably the easiest and most efficient way of doing so. Great force or intensity is probably not needed, scrubbing is not needed, provided your nails are short, it's just something to do while you're waiting out the 20 seconds.
Helps the water get to the detergent/lipid mix at the end so it can be washed away.