A "ground" is, generally speaking, where you want the electricity to go. Otherwise (and I'm grossly oversimplifying) it basically goes "everywhere".
If you wear a grounding strap, virtually all of the electricity goes into YOU. Which, when you're working with the extremely low amounts of electricity generated by static whilst you're working on a computer, is less harmful to you than it is to the components you're messing with (you literally won't even feel it most of the time, it can fry unprotected electronics).
However, in a situation where you're working with more power (something plugged into a wall, or even the unshielded capacitors inside a computer's power supply - which is why you're NEVER supposed to open them up), all that electricity going into YOU is a major problem - one that usually leads to a charbroiled technician.
So in a nutshell:
DO use grounding straps in situations where the voltages/amps are very low and dangers of shock are more concern to the machine than you. Know that you WILL take all the shocks.
DON'T use a grounding strap in situations where your body can't take the potential shock.
When in doubt, don't use a strap. Accidentally frying a component inside the computer isn't nearly as bad (or as expensive) as accidentally frying the technician.