The theory of the anti-static strap is that if you and the computer chassis are at different levels of electrical charge, or electrical potential, there is the possibility of generating static electricity. This charge flow can be damaging to CMOS type components even if there is no spark. As little as 2000v can damage a CMOS IC, whereas it takes about 20,000v to generate a spark (yes, even the little ones at the tips of your fingers).
Connecting a static strap to your wrist, and the other end to the chassis will provide a connected path between you and the chassis for the electrical charges to flow, and therefore equalize. This will help prevent damage to components. It's irrelevant if the computer is plugged in or not as far as this is concerned, but it's indeed hazardous to have the PS plugged in. Calling them "grounding straps" is really a confusing thing. You don't so much "ground" yourself as much as you equalize the electrical potential between you and the computer.
Just touching the chassis with your bare skin while working on the computer will indeed do the same thing as the strap, but there's always the possibility that you'll forget. Also, as you move your feet on the carpet, you'll be generating charge, so your potential changes constantly. Just touching the chassis briefly and then grabbing a board is not a guarantee of safety. You must be in contact the entire time.
Another little trick we used at my old shop was to put a few drops of liquid fabric softener in a spray bottle of water and spritz down the carpeting around the shop benches every morning. This reduced the tendency of the carpet to produce static charge and made the room smell spring fresh too!