How did the defeat of Carthage give rome control of the Mediterranean

The defeat of Carthage gave Rome control of the western basin of the Mediterranean, rather than the whole of this sea. Although Rome annexed mainland Greece in the same year as the destruction of Carthage, the further expansion into the eastern Mediterranean occurred later and was unrelated to the Punic Wars.

Rome took over the Carthaginian possessions in western Sicily in the First Punic War and seized Sardinia and Corsica from Carthage soon after that war.
Rome took over the Carthaginian territories in southern Spain in the Second Punic War. The Numidians of Algeria switched their alliance from Carthage to Rome.

Rome destroyed Carthage and took over her remaining territories (Tunisia and western Libya) in the Third Punic War.

Rome controlled the whole of Italy. Syracuse, the most powerful Greek city-state in eastern Sicily was a Roman ally. The Mediterranean coast of southern France was under the influence of the Greek city of Massalia (Marseilles) which was a long standing ally and friend of Rome.

This amounted to total control. In the first of the three wars, Rome also moved from hardly having a navy to speak of to building the largest fleet in the western Mediterranean.