I think the idea is that green tea will kill the bad bacteria in your mouth that causes cavities.
Not a bad idea, but I agree that the green tea craze goes a bit far. From what I understand, the flouride in tea isn't a whole lot more than the amount that's in your flouridated municipal water. I don't know how the extract compares to a cup of green tea in any of these factors, though. It could be just as good, or it may lose some of these properties.
I wouldn't expect the extract to stain your teeth, but I'm not exactly sure what's responsible for the stains, really.