The short answer is that this is a very complicated topic—much too big for an AH thread. It was through a whole series of factors that "Romanization" took place, varying by the place, the time, and the culture(s) involved. It would require an entire book to really capture this "answer," and in fact several prominent efforts have been made. Perhaps the most noteworthy is Greg Woolf, Becoming Roman. The Origins of Provincial Civilization in Gaul (Cambridge, 2000). Woolf, building on the older theses of others, argues persuasively that "Romanization" was largely a top-down process, meaning that the elite members of the non-Romanized society were compelled, induced, enticed, and ultimately rewarded for buying in to the Roman way of doing things.

In many (not all) provinces, Rome took steps to reshape the urban landscape. In Gaul, this meant accelerating the process (already in progress) of relocating urban centers out of the hilltop oppida and into the plains and river basins where they could be accessed by road and waterway, the arteries of the Roman economy. For the Romans, this meant more efficient extraction of resources and the elimination of strongholds which might later become obstacles to law and order. The role of the Gallic elite in this geographic reorganization was paramount, and they "led by example" for those over whom they held political and/or social influence.

In practical terms, this meant that these elite Gauls came to speak Latin, to take Romanized names, to spend their resources for improvements in the towns along Roman lines (such as funding temples to the divine emperors, dedicatory statues in Roman style, basilicas, roads, aqueducts, etc). The iconography on their tombs became Roman, or at least came to incorporate Roman thematic elements, and their families became involved in Roman bureaucratic machinery, with their sons going off to be educated as Roman elites, in the Provincial capitals or even in Rome, to serve as tribunes of the soldiers, or apprentice with prominent senators, and thus to begin the climb towards higher office in the Roman state. It took time and a little luck, but eventually we see participation of "provincials" at all levels. Septimius Severus was born in Leptis Magna, in Libya, and while his mother's family was old Italian, his father's side were provincial and old Punic. He rose to become emperor (he is just the classic and easiest example, but there are many more examples of "provincial" families making their way in the Roman system via inscriptions).

The Roman economy exerted enormous passive economic pressure on provinces to conform. Roman business interests privileged advantageous arrangements within the provinces at all levels, and Roman economic activity fueled astounding rates of urbanization, especially in the less urbanized West (Britain, Gaul, the Danube corridor, much of Spain, the North African Maghreb, etc). As I mentioned before, access to and extraction of resources drove urbanization in many cases, with new towns springing up to serve these demands (much like the pop-up towns in the oil-booming regions of the American midwest), and with them the roads, waterworks, and other infrastructure. Provincial denizens who conformed to these emerging urbanism patterns benefitted from passive integration into the nascent system, no matter if they sold carrots or organized mining contracts, and provincial individuals or groups who chose not to participate in these emerging networks were, quite literally, sidelined. The economic machinery of Romanization is itself another huge topic and I don't have the knack nor the knowledge to do it justice here, but economic processes are both complementary to Woolf and also partially in opposition to his thesis, in my opinion.

As you mention initially, colonization played a role. I think probably the greatest role of Roman colonies in the provinces was, however, as anchors or "seeds" of those initial urbanizing pushes. The vast majority of Roman colonization happened under Augustus, when hundreds of thousands of soldiers under arms were settled throughout the Med. This was a huge "Romanizing" horizon which led to secondary urbanization, again especially in the West, in subsequent centuries.

Professor Carlos Noreña at Berkeley does work on Romanization, especially as it relates to Roman State Power as a driving force (a slight divergence from a model of economic impetus for Romanization/urbanization), for which see (e.g.) "Romanization in the Middle of Nowhere: The Case of Segobriga," Fragments 8 (2019), 1-32.

Standard texts for this topic are Woolf (above); Millet, Romanization of Britain (Cambridge 1990); Lee, Romanization in palestine : a study of urban development from Herod the Great to AD 70 (Oxford 2003); Churchin, The romanization of central Spain : complexity, diversity, and change in a provincial hinterland (Routledge 2004); Ørsted, * Roman imperial economy and Romanization : a study in Roman imperial administration and the public lease system in the Danubian provinces from the first to the third century A.D. (1985); MacMullen, Romanization in the Time of Augustus (Yale, 2000); etc.