Fatter tyres are generally run at a lower pressure, a lower tyre pressure means a larger contact area with the ground, a larger contact area means a lower ground pressure, a lower ground pressure means you'll give the 'dart' less help to get through. ergo, fatter tyres get less flats.

All things being equal, if you run a fatter tyre at the same pressure as a skinny, you'll get more flats: the fatter tyre hits more road, i.e. the contact area is the same, but there is more 'rest of tyre' in proximity to debris on the ground.

Yet, a fatter tyre generally has more rubber too, so it is harder for a dart to get through.