I've been traveling through Sicily for a few weeks, and I've noticed a very interesting blend of architecture here. It seems like the area has changed hands many times--for example, the Duomo at Monreale has a beautiful interior representing several periods of art and architecture from the entire Mediterranean (even Islamic architecture). There are several volcanic regions, including one (Stromboli) that is continuously active and erupting. The flora and fauna are vastly different from mainland Italy as well. I'm curious to know what attitudes towards Sicily were in Roman times, either popular or of rulers.
Sicily has a big significance in ancient history because it has belonged to three major civilizations: Greece, Carthage and finally Rome. It was settled by the Greeks around the 700s BC, who founded the city-state of Syracuse. The Greek culture of the island eventually spread into mainland Italy, which is where the Romans first made communication with the Greeks. During the early days of the Republic, Sicily was not part of Rome but it was very important as a site of communication between Rome, Greece, and Carthage who owned most of the western part of the island. During the Pyrrhic War between Rome and Greece, Rome gained control of most of Sicily from the Greeks, although a small part still belonged to Carthage.
This territorial dispute, among others, eventually led to the Punic Wars, in which Rome and Carthage went head to head over who would be the most powerful empire in the ancient Mediterranean. (The Second Punic War is probably one of the most significant in Roman history, it's the one where Hannibal crosses the Alps with elephants!) Eventually, the Romans won, and officially made Sicily a Roman province. After that, the island became very important to Rome for trade. Its rich soil provided grain for the city of Rome, and its central location allowed the Romans to make further communication and conquest into Egypt and Northern Africa.
Part of the blend of architecture you refer to is because of Sicily's unique place in many different cultures. Starting out as a Greek stronghold, the culture remained strongly Greek even after Roman annexation (to the extent which most people living there spoke Greek instead of Latin). But it also has had a strong mix of cultural influence even after the fall of Western Rome, being a part of the Byzantine, Arab, and Norman empires throughout the Middle Ages.
TL:DR Sicily was very significant to the Romans, especially before their expansion into Egypt and North Africa. It was an important political ally, then a source of income and trade. In addition, Sicily was one of the first territorial acquisitions by the Romans outside of mainland Italy, signaling the beginning of their imperial ambitions.