when did the people in italy stop speaking latin, and start speaking italian?
First things first: people in Italy did not start speaking what we know today as Italian until the XX century. As the Italian philosopher Luciano de Crescenzo puts it "Italian is not the language of Dante, but the language of Pippo Baudo". Until the expansion of the radio, and more particularly of television, people nostly spoke their regional dialects or languages (Tuscan, Lombard, Venetian, Neapolitan, Ligur, etc), and these dialects are still rather widely spoken, although they are more prevalent amongst the elder folks.
Having said that, the gist of your question concerns the emergence of Romance languages as separate entities from Latin. That can be traced more or less to the 600-800 AD period, as attested by some documents like the canons of the Council of Tours (AD 813). For this, I link you to an answer I gave yesterday concerning the emergence of Romance languages, which based on the Council of Tours, the Oaths of Strasburg (AD 842), the glosses of Reichenau (late VIII century), and Saint Isidore of Seville (early VII century). I hope it helps: