Rome and Carthage clashed in the Punic Wars because they were the two greatest superpowers of the Mediterranean world. When their empires came in contact with one another, they fought.
For many years, Rome and Carthage coexisted well enough. As time went by, however, Rome began to worry about the spread of Carthaginian power. Then, in 264 BC, Carthage and Rome became involved in Sicily. A war had broken out between two cities on Sicily. This war came to involve both Rome and Carthage. This was the start of the Punic Wars.
Overall, the Punic Wars came about because these two powers had gained large spheres of influence that started to come into contact. When this happened, conflict was almost inevitable.
Both Rome and Carthage were two rapidly expanding powers, and with both nations expanding in the same direction, they eventually clashed and created a conflict known as the Punic Wars.