Why did Rome's victory in the Punic Wars make it so powerful in the world in general?
First, we must realize that Rome was already quite powerful even before it won the Punic Wars. During the Punic Wars, it was also fighting in Syria against the Hellenistic Seleucid Empire and in Macedonia against the Seleucids and other Greeks. Before the Punic Wars ended, Rome had won these wars. This shows that Roman power was extensive and was increasing.
The victory in the Punic Wars extended Roman domination to cover much more of the Mediterranean World. Carthage had been the naval power in the area, particularly in the west. By defeating Carthage, Rome became the undisputed leading power in the Mediterranean. This allowed it, for example, to start to expand into what is now Spain and to control Northern Africa.
Before the end of the Punic Wars, Rome controlled much of the Italian peninsula and some areas to the east. The wars destroyed their only major rival for power to the south and the west. This meant that Rome had no real rivals as a "superpower" in their world.