The meaning of French Style Green Beans is just that the French Style green beans are slimmer and grow thinner than regular green beans.
So it's just a French Style Variety of the green beans that derive from French Locations whereas American grown green beans are more thicker and not as thin.
I actually prefer the French Style Green beans myself as they're easier for me to chew and for some reason they tend to taste better than regular American Style green beans.
But that could just be me and my tastes.