This is an excellent question which is very relevant to those who study this period in Greek history/archaeology, and therefore it is not easy to answer. I will give you my take on it, but understand that a great deal of debate exists on this topic and my answer is not gospel.

For the purposes of this, let's stick to the earliest horizon of Greek "colonization," from about the middle of the 8th century to about the middle of the 7th century. The 8th century in the Mediterranean was what John Boardman called the "first really busy period," when Greeks began to push beyond the Aegean for the first time since the end of the Bronze Age, some three centuries earlier. With them at around the same time are Phoenician ventures headed west to North Africa, Sardinia, and southern Spain. The Etruscans, too, were plying the local waters of north-western Italia in this same period. By 750 BCE, the Greeks and Phoenicians are both visiting Sicily and Italia, and the Phoenicians are pushing further west. By 700 the Phoenicians are firmly established at Carthage, on Sardinia, at Gadir/Cadiz, and in Andalusia, though the sites of that last category will be gone by 650 (see here). Also by 700, the Greeks had established several "colonies" in the west, mostly on the eastern end of Sicily (like at Syracuse or Megara Hyblaia) and in southern Italia (at sites like Metaponto or, a little later, Tarentum).

Greek sources, writing centuries later, often refer to these early settlements as true colonies, with an organizing party and an expedition led by an oikistes ("founder"). We hear of the notion of mother cities sending out "colonies." These stories are, functionally, somewhere between fact and myth and are very elusive. Personally, I think they are later-day conceptions which were grafted onto an unknown past--in other words, inventions. It is a tangled, complicated subject because the act of creating new Greek "cities" was ongoing for centuries, an organically-evolving process of moving and settling human beings. The first "settlers" to reach Sicily in the second half of the 8th century BCE might have been led by charismatic individuals, and they may have involved meticulously-organized parties, but I think the process was much different than it would have looked two or three centuries later.

"Mother Cities" : it is important to get a handle on what the Greek world was like in 750 BCE. It is easy to imagine situations in the void which did not actually exist "on the ground" at the time. First, the idea of a city, a polis, at all. In 750, places like Athens, Corinth, and Argos were still in the fledgling stages of state formation. The inhabitants of these areas were still sorting out the basics, like the notion of removing burial of the dead to the edges of occupied space, of working collectively to solve problems shared in common, of erecting shared buildings for worship and public business, of defining shared territory, and of establishing identity as "Corinthians" vs "Athenians" vs "Argives." To call Corinth a "city" or a polis in 750 is pushing it; to then imagine "Corinth" getting together to send out a party of settlers to distant Sicily seems a little far-fetched. A common litany in older scholarship is that the Greeks went west because of population pressures at home. It is true that what limited evidence we have points to expanding populations almost everywhere in the 8th century world, but archaeology has also shown that population pressure in Attica, or the Corinthia, was (apparently) directed towards local unoccupied sites, arable or semi-arable plots in the fertile valleys which had been only sporadically occupied since the collapse of the Mycenaean world.

If not state-sponsored "colonization," then what? Robin Osborne, in 1998, wrote a brilliant article on this subject titled "The Nature of Greek Settlement in the West." In it, he argued that we should think carefully about early colonization and question later, classical paradigms for how "colonies" get founded. He examines the archaeological remains of 8th century Greek settlements in the west, in particular noting that the "colonies" (like Megara Hyblaia) look nothing like the "mother cities" as far as archaeological remains (pottery, e.g.). He envisioned a "mobile population" of sailors, Greek and Phoenician and Etruscan and all the rest, moving around in a world which had become more open, more interconnected, more "busy" by degree since the "Dark Ages" after the end of the Bronze Age. They weren't necessarily "Corinthian" or "Athenian" and probably would not have possessed a clear conception of what such terms meant. They were, instead, "sailors," "traders," explorers." If they came from the Corinthia, perhaps they carried Early Protocorinthian pottery with them, but they also fancied Attic amphorae, Argive kraters, Euboean-style cups, and unlike their land-lubber cousins back home, they were plugged into emerging networks which gave them ready access to such variety. The graves they left behind in their new homes on Sicily support this. At Pithekoussai, some of them drank wine from local imitations of Euboean cups, mixed the wine in Attic vessels, stored their oil in Corinthian aryballoi, and wore Phoenicio-Syrian amulets around their necks. To what extent then were they "Corinthian" or "Athenian?" I highly recommend this Osborne article to anyone interested in this period. In his conclusion he describes this "mobile population," suggesting it "was comprised of individuals and small groups out for their own gain, [who] from time to time came to believe that more or less permanent settlement on foreign shores was both in their immediate best interests and was sustainable."

As an addendum: I think there are different motivating factors at work here, all lumped together in this "mobile population." Ships are expensive, and were incomprehensibly expensive in 750 BCE. I suspect the Greek visitors to the West in the 8th century were rich, to say the least, and probably the remnants of a Bronze Age merchant class which had become severely depleted in the intervening centuries but managed to sustain itself on what limited trading opportunities existed. We find exotic goods all over the place in Geometric Greek graves from the 10th century all the way to the end of the 8th century, and there is no need to suppose that all of those objects were brought to Greece by "Phoenician traders." I further suspect that these families, intimately tied to the seas for their identity and livelihood, were the big losers in the slow process of polis formation, when land tenure (owning land) became the defining feature of elite status and membership in the "city." By the end of the 8th century we find groups all over the Greek world laying claim to long-forgotten Mycenean tombs in the countryside, announcing (inventing) that the dusty bones inside belonged to their ancestors (they didn't), and that they therefore had claim to ownership of that land. Owning land was a novel concept, and there was a full-blown land grab frenzy. This hypothetical merchant class, or what was left of it, if they were the losers in this trend, then would have strong incentive to seek new places to put down roots in the West. This is, of course, assuming that they wanted to participate in the changing competition over the definition of elite status, and needed land to belong to the emerging land-owning aristocratic clique.