August is Italy's peak holiday month, and the closures are not a rumour: private businesses and public offices shut for at least a couple of weeks under the chiusura estiva. Anyone living here is better off finishing administrative business by the end of July, because banks, lawyers, notaries and bureaucrats of every kind will be away for most of the month, if not all of it.
Visitors run into the softer version of the same thing. Restaurants and shops close without notice, and small family run businesses in particular tend not to update their August hours online, so the opening times on a map are frequently fiction.
The centre of gravity is 15 August, Ferragosto. The holiday has Roman and Catholic roots and now functions as the official height of the season: the country drops to minimal operations, cities empty and the coast and the mountains fill. Expect heavy traffic in the days either side, and reduced public transport.
One useful distinction for the middle of the month: state run museums and archaeological sites are usually open on Ferragosto itself, while privately run galleries may not be. If you are staying in a city, the public institutions are the ones that will still let you in.