Summary
The first Punic war started with disputes between Messina and Syracuse soon becoming a war between Rome and Carthage which resulted in Rome taking over Carthage land (Sicily). In the second Punic war Rome took over Corsica and Sardinia. Hannibal took his military of 90,000 infantry, 12,000 cavalry, and some elephants and marched from already conquered Spain to the Alps down into Italy. He surrounded the Roman army by surprise but lost because that army was two times its number of men. The third Punic war was the point where Carthage was no longer a threat to Rome but with some convincing from the Senate, Rome went on to destroy and burn Carthage into the ground in North Africa. Enslaving all survivors and leaving no trace of Carthage left. After Hannibal died in the 3rd Punic War, no one seriously challenged Roman power for close to 6 centuries. Therefore, Rome continued to expand and became the supreme empire we know of today. The Roman Empire was from the Atlantic coast of Spain to the boarder between Greece and the Asia Minor, also known as modern day Turkey.
After Carthage was destroyed, it was a power vacuum with no one leader that Rome easily took over. Rome was then able to use all of the Carthaginians infrastructure and continue to expand into Spain and the Asia Minor as shown above.