The catapults were first created in ancient Greece according to SCITECH as found in Arizona and we think it was around the 4th century BCE during the early Hellenistic time.  So the reason Greece originally made catapults was because of the rise of cities with walls. According to Brick, as found in Greek Boston.

They started to become more popular like Athens, Syracuse and later Macedon were surrounded with huge walls and Greece wanted an easy way to get to their enemies if they needed to if something like a war would have happened. 

But when they created them they figured out this thing called a torsion spring which really helped the Greeks improve in a lot of things. Like they were able to make catapults bigger and stronger catapults with these springs. Like heavier arrows and boulders could be launched, they could launch them longer because of the better force, because of spring it had stronger impact against walls and they used boulder catapults with big torsion springs to break down walls.