Damnit, I should be studying. I blame you for this.

So a group of mercenaries in Sicily (calling themselves "Mamertines" after an Oscan god) had been let into the city of Messana which they took over.

So the tyrant of Syracuse (Hiero) wasn't terribly fond of this and began attacking. After suffering a setback the Mamertines appealed to Carthage (which controlled the western half of Sicily) and the two defeated Hiero. Then when the Carthaginians wouldn't leave, the Mamertines appealed to Rome for assistance.

Rome was reluctant because this was outside of Italy but many elites recognized that Messana was crazy wealthy or they saw an opportunity for conflict and glory (this latter issue was a constant problem for Rome). The Romans then threw out the Carthaginians which triggered the war. It went from 264 - 241 BCE with Rome slowly whittling away the Carthaginian-held cities on Sicily while Carthage originally dominated the sea. Eventually Rome got the upper hand at sea and invaded Africa causing Carthage to sue for peace.

So it was a war that neither had really planned for. Both knew that Messana controlled the easy trade route to Sicily which allowed one to tax ships passing through. Opportunism led to their interference which then triggered a long, bloody war. Sicily was very important to Carthage and Rome never accepted anything short of total victory. So both sides fought to the bitter end.

If you believe the traditional narrative, which scholars sure as hell don't, the rivalry traces back to Aeneas, the Trojan who establishes the royal line of kings from which Romulus and Remus would spawn from, and Dido, founder of Carthage. Livy writes that Aeneas, side-tracked by the petty pantheon of gods and goddesses, lands at the newly built Carthage with his band of surviving Trojans. Dido and Aeneas fall in love and spend many years together, however Jupiter and Venus eventually remind Aeneas of his greater destiny and force him to abandon Dido. Dido, upon learning this, curses the descendants of Aeneas to eternal strife with the Carthaginians and burns herself alive upon a pyre.

In reality, the Romans and Carthaginians had an amicable diplomatic relationship for the most part, with historical evidence surviving for treaties between the two which establish their spheres of influence in Italy (when Rome was still a meddling central-Italian power). However, after the Samnite Wars and the conquering of Magna Graecia along with the defeat of Pyrrhus, war between the two powers became more or less inevitable. Rome had a hunger to expand and Carthage wanted to retain ownership of Sicily. The Mamertines' seizing of Messana (and subsequent (and contradictory) call for aids to both Carthage and Rome) was just the spark.

Sources:

Livy's History of Rome.

Adrian Goldsworthy's The Punic Wars.

Read Carthage Must Be Destroyed by Richard Miles