Why is Italy famous for coffee when they don’t grow any in Italy ?

According to me, pizza is more famous than coffee in Italy. Because pizza really represents my country for its beauty and goodness; coffee was famous until ten/twenty years ago. 

Actually, unless someone is growing coffee as an indoor plant, no coffee is grown in Italy at all. Italy's coffee fame rests on its coffee companies abilities as roasters and blenders for espresso

Green beans or red ripe beans does not give you coffee.  This is a controlled process of roasting which Italians have perfected to their standards. Mostly Cappuccino.

   The Turkish Muslims introduced coffee to Venice traders, so it was first established there. 

Because the Italians love drinking coffee which is of a very high standard.

there is more to coffee than growing it its how you prepare it  in order to drink and enjoy it

In Italy, they like coffee, so when they often drink coffee before working, after working, breakfast, etc.

And also, cafe seems popular place for information exchange.

In Italy coffee is culture. You have so many varieties and they are quite picky about it. I don't remember ever going into an Italian home and not being asked if I'd like a coffee, or as they call it caffe. It's basically their lunch break, breakfast, after any meal and anytime really. Its good to notice the difference that you will rarely find it in any other country. In Italy at the bar you get a coffee, standing (un caffe al volo), which literally means a coffee on flight, cause you just get it, and out you go. They charge you extra if you sit at the bar.

Because they drink a heavily concentrated  form of it,  served in little sippy cups. 

They laugh at Americans with their BIG GULP  cups,  and think they are drinking  basically  muddy water. 

What Italians drink is closer to pure coffee syrup almost. They don't need those 

 American bladder buster cups. 

You just have to taste it. Americans try to make something called gelato, too, but it bears no resemblance to what you get in Italy.

Because they have their own unique ways of preparing it.