Edit: Some grammar changes.
When the Italian Air Force was first created in 1923, it was obvious to everyone in Italy's military establishment that land-based aviation could more or less cover all of Italy's strategic interest, and so calls to build an Italian carrier were put on hold. Up to 1938, even the Supermarina (Italy's Naval Headquarters) was in agreement with this. France's battleship fleet was being modernised and Italy hadn't the resources to build both carriers and the heavy units to compete with the French.
In November 1940, the British carrier Illustrious delivered a devastating and unforeseen attack on the main Italian naval facility at Taranto (although it wasn't the first time British fleet air arm had struck the Italian navy in port), sinking a battleship and heavily damaging two others. A few months later, the Regia Marina was routed by the British Mediterranean Fleet at Cape Matapan. The Italians were aware the British carrier Formidable was in the area but decided to attack anyway. During the day, Formidable's air group caused the Italian squadron considerable distress. An Italian cruiser squadron gathered around the cruiser Pola which had been crippled by an air attack; during the night, the Pola was found by British radar, and the radar-less Italian capital units were torn to shreds by the British battleships. Four Italian cruisers were completely lost. Three British sailors were killed.
It now became obvious that Italy had to produce a fleet carrier that could keep pace with her capital units and provide air cover for the fleet and air attacks on their opponents. The Italian high command, and Mussolini, was now unified behind the idea of an Italian aircraft carrier. Traumatised by Cape Matapan, Italy's heavy units never again went into the eastern Mediterranean and never again tried to fight the British without land-based fighter support.
The 32,000 tonne trans-atlantic liners Roma and Augustus had already been put into conversion to become auxiliary carriers. Now the Italians decided to turn them into real fast fleet carriers. The shocking state of Italian industry prevented this and neither ship became a functional aircraft carrier before the end of WWII.
Italy did not have a fully functional aircraft carrier until Guiseppe Garibaldi was commissioned in 1985. A little bit of background on this ship. Throughout the cold war the Italian navy's task was the surveillance and monitoring of the Mediterranean and potential sea control of that area, potentially helping the US Navy prevent a Soviet breakout from the Black Sea. By the early 1960s the Italian Navy was going through a reforming period. Its strategic tasks were to control the chokepoints of the Mediterranean. However, by the 70s the Italian fleet was aging badly and something had to be done, so the Naval Law of 1975 allowed Italian companies to market and export warships abroad. Here was born the European shipbuilding giant Fincantieri. This commercial move allowed Italian shipbuilding to expand despite a lack of domestic orders; those orders came anyway in 1977. Guiseppe Garibaldi was ordered in 1981 to fulfill Italy's strategic naval operations and finally acquired fixed wing Harriers in 1989. The Italian dream of an aircraft carrier had been fulfilled.
Italy's carriers - today the Guiseppe Garibaldi and its newer, larger sister, the Cavour, are not designed to project power abroad like a US supercarrier. They weigh under 30,000 tonnes (less than 1/3rd of a US carrier) and carry a small number of less-capable STOVL/VTOL aircraft compared to the big American multirole carrier aircraft, and no airborne early warning planes. The Garibaldi's deployments until Libya was the Italian naval patrols named Golfo 1 and Golfo 2 in - you guessed it - the Persian gulf. In this way Italy was able to contribute to NATO combined operations. Cavour was acquired to expand Italy's capabilities in this regard.
Carriers are not just for power projection. We have seen that in the past, a carrier on-station allows rapid air support to naval units, which could be decisive. Italy's strategic situation in the cold war resembles Britain's; the control of naval chokepoints (in Britain's case the GIUK gap; Greenland Iceland UK area over which the Royal Navy had to prevent Soviet submarines from crossing). Smaller carriers known as ASW (antisubmarine warfare) or just helicopter carriers are beneficial in that they can carry a large number of helicopters. Guiseppe Garibaldi can carry up to 18 helicopters compared to destroyers and frigates which might be able to carry just 1 or 2. Helicopters are the strongest weapon against submarines because of their speed, range, and virtual invincibility. By using dipping sonar and torpedoes, helicopters can strike submarines directly without being attacked themselves. The Italian navy acquired Harriers to use as fleet defence aircraft to protect squadrons at sea from air attack, with actual strike operations as a secondary priority.
In addition, the large size of an aircraft carrier allows it to be used in humanitarian operations, flying the flag and diplomatic visits, and as a show and sign of a country's strength and prestige. Force projection does not mean invading entire countries, but also participation in NATO combined operations around the globe as part of Italy's military responsibilities. Carriers also act as a marketing tool for the national shipbuilding industry. The Italians are no less interested in this than the British and the French. That their carriers turned out to be more beautiful is both in keeping with the national stereotype of design, and the long tradition of Italian military shipbuilding.
Sources:
The Units of the Italian Navy (1997) - Imago
The Italian Navy in World War II (1957) - Marc Antonio Bragadin, US Naval Institute