Like other areas after the fall of the Western Roman Empire, Italy came under the control of German tribes that soon set up their own kingdoms. What caused Italy to fragment politically? Why were city state republics set up there and not in other countries?
Why Italy didn't unify might just have as many reasons as why the Roman Empire collapsed. Fundamentally, strong local institutions were never superseded by a large overarching "national-style" authority as happened elsewhere. Landmarks in disunity include but are not limited to the following, vaguely chronological, list of events:
Highly autonomous urban communities are not unique to Italy. In most post-roman cities we could expect to find largely homogeneous political and governmental institutions: a council, of some form or other, would typically exist tracing its origins to the Senatorial-style local governments exported by the Roman Empire. Leading this council would typically be the local Bishop or religious head, another holdover from the roman period where political and religious authority was not differentiated. However, over time in most of Europe urban governments were co-opted by monarchial authority, as the "political" role of bishops was replaced by a monarch, monarch's representative, or a locally elected representative acknowledged by the monarch. In Italy this didn't happen for a really long time: not only were these these council mechanisms particularly strong, but no permanent stronger top-down power emerged.
But if, reconciling my previous two paragraphs, the highly autonomous Italian cities nonetheless existed within the confines of a unitary state up until the Carolingian conquest, was it Charlemagne's empire that affirmed fragmented government? Probably not, since however unstable Italy's carolingian monarchies might have been they were not a particularly unique or problematic part of the Empire. Italy was about just as politically unstable — by design — as the rest of Carolingian Europe (and for what it's worth the last person to rule all three parts of that empire had risen from the very post of King of Italy).
In fact, Italy's role in the Carolingian Empire was vital to its ability to call itself "Empire" at all. The same legal and political constructs dating from the Roman era which allowed Italy's cities to successfully govern themselves also allowed the Italian leadership adopt the framework of empire (Carolingian first, and Holy Roman later) on top of the existing body of Roman Law. So it would be the Italian leadership (above all others the clergy) to legitimize the claims to the title of Emperor on behalf of ambitious rulers. But just as German Emperors partially developed their political authority on claims of continuity with the old Roman Empire, so did they increasingly find it necessary to make concessions to the highly autonomous Italian urban communities who anointed them. By the time Germanic emperors were secure enough in their appropriation of the notions of "Empire" to attempt to centralize authority in Italy, so too were the Italian cities able to defend their autonomy by force of arms.
And, as an aside, we ask why Italy was not unified, but we could also ask the same question of Germany. The german Reich adopted the Roman notion of "Empire" with little to show for in terms of political unity. Thus is the turning point for Italian fragmentation Emperor Otto's crowning as Holy Roman Empire in Rome?
The answer is "Sort of." The old chestnut circulating in Italian classical liberal (and thus anticlerical) circles of 19th and early 20th century academia attributed disunity to the power of the church, especially the Papacy. Starting from Otto's coronation, the Papacy's insistence on anointing an "Emperor" (a practice beginning with Charlemagne, but really enshrining itself with the Ottonians) eliminated the legal or social space for a monarch claiming the right to rule Italy. Further, the Papacy's insistence on interfering in European affairs to safeguard its own power (leading, with other factors, to armed conflicts on the peninsula) would amplify the regressive effects on Italian society.
But did the presence and influence of Europe's religious head on the peninsula really place a ceiling on the the unity of the country, or did things go the other way 'round? The periodic conflicts which later emerged between Papacy and Empire saw most Italian urban communities consistently rally behind the cause of the Church and its primacy over the Empire (a primacy later extended to all of Europe's monarchs, creating a strange and ignored paradox given that as per the classical Roman interpretation, the Emperor ought to be only temporarily embarrassed to not be ruling the known world). And besides, for all the papacy's scheming (and the Italian urban communities' backing) the Catholic Church really could not stop the slow but unmistakable cession of institutional power to secular institutions.
So a more complete summary would look at the fact that the anointment of an "Emperor" was a consequence of no comparable figure emerging in Italy, independent of the Papacy's ambition. Why did no such figure emerge? Initially, because the post-Roman period did involve a fair bit of turmoil in the peninsula, leaving political actors weak and ineffective. But no figure able to make bid to unite the peninsula emerged even as the various Italian city-based political apparati consolidated into larger states which, although small, in the 14th and 15th century nonetheless appeared to be some of the more efficient in Europe. Even as individual dynasties and magnates subverted the old council systems and elevated themselves to positions of near-absolute power (the Signorie, who did their best to emulate the monarchs that existed in the rest of Europe) none was able to garner the resources to even begin to consider a serious bid to sieze the whole peninsula.
In part, this is because the collegiate nature of the Italian urban governments (even under the Signorie) was useful to foster inclusiveness and prosperity, but not very good at organizing resources for conflict. But there was another, broader issue: if in the rest of Europe, one group (or subgroup) was able to appropriate the structures of the old Roman provinces to create the embryos of what would become France, Spain and England, which group could have done this in Italy? In Italy, the superstructure of government was not appropriated by any one group. Instead, it disappeared and was not reinstated (remember the Gothic and Lombard Wars?) thus leaving behind only local government. By the time this structure began getting built up again by the Signorie it was too late: the Italian states had not only become players in European politics able to call upon larger and more powerful allies, but the larger and more developed states of Europe were beginning to turn to the fragmented but prosperous peninsula as a land to exploit and conquer.
I also have a few older answers where I narrate the nitty-gritty of disunity, this absurdly long answer is one of them.