If you hold a U.S. passport and an onward flight ticket you just may pass through Italian Immigration and get an entry stamp on your passport. For your connecting flight AZ832 you need to get to terminal T3 for departure. To get into Rome you can use the Fiumicino Express train to Roma Termini train station which takes 30 minutes non-stop at a fare of €14.00 or chose the Sabina-Fiumicino line (FM1) which stops in some of the main Roman railway stations (Tiburtina, Tuscolana, Ostiense, Trastevere) and costs €8.00 one way.
Also passport holders of all EU countries and of Andorra, Antigua and Barbuda, Argentina, Australia, Bahamas, Barbados, Bermuda, Brazil, Brunei, Canada, Chile, Costa Rica, Croatia, El Salvador, FYROMacedonia*, Guatemala, Honduras, Hong Kong (SAR), Iceland, Israel, Japan, Liechtenstein, Macao (SAR), Malaysia, Mauritius, Mexico, Monaco, Montenegro*, New Zealand, Nicaragua, Norway, Panama, Paraguay, Saint Kitts and Nevis, San Marino, Serbia*, Seychelles, Singapore, South Korea, Uruguay, Vatican City, and Venezuela can explore the city during the layover [* = e-passport required]. All others either need to stay in the international transit area or obtain a tourist visa [€60] from the Italian Embassy of their country prior to travel.