For the Greeks (I can't speak as to the Phoenicians), overpopulation was likely a concern. Greece does not have copious amounts of arable land (although some parts are quite fertile) and disputes over farm land could be vicious. For instance, the poet Hesiod is known to have been in a very bitter dispute with his brother over inheritance of their father's farm.

Having men go out and found new colonies is a practical solution to that. And it was men, generally. The Greeks would usually intermarry with local populations.

Not all of their neighbors were too strong. When Sparta needed new land they conquered their neighbors and turned them into slaves. They didn't set up new colonies. And that's just the most famous example. Of course, we all know there was war among the Greek city-states. But that wasn't a sustainable, practical solution for everyone.

Trade and commerce was likely a huge reason for both Greeks and Phoenicians. With Greeks the new city would be independently governed. But they did retain economic links with their "mother" city. Having a close cultural relationship with a city on an important trade route can be key.

Greek colonies served as everything from trading outposts (with obvious economic benefits to the home city) to escape valves for dumping politically-dissatisfied populations (which could stabilize the community). They existed to build up the strength of the home city, not just to rid itself of excess population.