They are often used in conjunction with a dissipative mat that is made of a conductive material with fairly high resistance to ensure that any charges dissipate slowly.
The straps should have a 1 megaohm or so resistor in, that's a safety feature so that you don't fry yourself if you accidentally touch something live when working with low voltages. They are not supposed to be used if working near high voltages e.g. anything over mains.
Leaving a PC plugged in and switching it off using the switch on the back of the PSU is a fairly common practice, I was taught to do it when I worked in a computer factory years ago. It does seem a bit dubious though.