Rome had decided to battle Carthage for Sicily since they were rivals
Carthage went to war with Rome as they were rivals and Rome was trying to take Sicily
The first Punic war was mainly on the water. Carthage had a better navy, but Rome turned the tables by boarding their ships using a gangplank. Rome won.
The second Punic War is the most famous, as a general named Hannibal tried to attack Rome from behind using elephants. Although they had crossed the alps, Rome had weakened his army a lot and Hannibal had to retreat.
In the third Punic War, Rome decided to invade Carthage and decimate it. Anybody who survived was sold into slavery. Rome then put salt in the ground so nothing would ever grow there again.
As a result, Rome is able to expand to other places and grow.