The way it works depends a little bit on the ride configuration, but basically one parent waits with the kid(s) while the other parent rides, and then the waiting parent gets to get right on the ride without waiting in line when the first rider is done (and then that parent waits with the kids). So the drawback is that the parents don't get to ride together, but it still beats not riding at all, or waiting in the line twice.
At many rides, the parent and kids wait at the exit of the ride. At Mission Space and Splash Mountain, for example, there are kids' play areas at the exits so the kids can have something to do while they are waiting for the grown-ups. You just tell the CM at the gate that you want to do the kid switch and they will give you ride-specific instructions, and often even physically walk you to where you need to be. It's a great service to people traveling with kids. IMHO that's what Disney does best: making it a vacation for EVERYBODY, not just one age group or party type.
Also, if you have another adult with you, they will usually let the third adult ride twice.