There are two parts to this question: how and why did the "Urban Republic" form of government emerge (and why it was so prominent in Italy) and what contributed to Venice becoming a leading political and economic actor in Italy and the wider Mediterranean. I've touched on both these topics in the past, although I'll be happy to answer any additional questions you might have, and there's always moe that can be said on any topic.

You might be interested in this discussion from about a few years ago which looks at some of the determinants which contributed to Venice's early success as a political and economic community, with links to a few more answers, including this lengthy narration from three years ago on Venice's earliest economic and political expansion. If you'd like to read more, the main introductory text on Venice I recommend to everyone is John Julius Norwich's A History of Venice, which narrates the life of the Republic from it's emergence under nebulous circumstances in the eighth century to its demise at the end of the eighteenth.

What you will see from the answers I linked above is that power in politics, especially politics between polities, does not materialize in ready-made form. A political actor's capacity to act and project power (be it of any kind, economic, military, or what have you) is a function of its institutions, which in Venice (as in many other places) where the result of slow institutional evolution whereby the ability to efficiently project political, economic, and even military power developed over time. Indeed while the seafaring community of the Northeastern Lagoons were generally recognized as prosperous very early in their history (serving, as they were, as a bridge between the prosperous East and unstable West starting from the seventh or sixth century) in the early centuries of their history they were nonetheless sparsely populated, politically inefficient, and beset by crippling internal conflicts. However, the melange of sparsely-populated island communities of the early lagoons were ultimately able to overcome their differences, develop effective institutions, and guard against enemies from both the East and the West (you can read more about that here. By the Twelfth Century, it was the Venetians to whom the German Kaiser and the Italian Cities would turn to in order to broker a peace after decades of conflict); complicit was Venice's growing political influence, but equally complicit was the city's ability to foster a reputation as belonging to both East and West while standing apart from both (the aforementioned Norwich refers to Venice as "an Eastern Capital" inexplicably transmuted to Italy's coast, and the Venetians were particularly good at convincing their Italian peers of this fanciful perception even if they were more involved in Italian affairs than they would admit). Of course, it is clear that after the Fourth Crusade the Venetians in part formalize, in part accelerate their achieving of a central role in Mediterranean affairs, culminating political and economic apogee in the 15th century, however my point is that the process by which this was achieved was slow and largely the result of successfully building effective institutions (which were, you will again see, not completely perfect nor impervious to their own decay, but they were able nonetheless to chart a centuries-long ascendancy before setting on a trajectory of decline).

You also allude to the (correct) fact that Venice was one of many Urban Republics in Italy, and this is also a topic worth examining. In the past, I answered this question by offering rambling narrations of the long-winded history of conflict among the Italian leadership, pointing out milestones towards disunity and aversion to unified rule. While fun for me to write, I think those rambling answers might have lost some readers along the way (if you really want to, you can read one of those here). I will be more succinct here: Suffice it to say that in Italy's political development, there emerged no central monarchial figure to supplant the Urban Bishops which, in the Roman fashion, acted as joint civil and secular leaders in their communities. Thus the city became the main building block of political society, and in these highly autonomous urban communities 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. And while in the rest of Europe monarchies emerged lashing together institutions, in Italy this didn't happen. Or rather, it happened very late: In the 13th and 14th centuries political dynasties did emerge that would subvert the old council systems and elevate themselves to positions of near-absolute power, however even they were largely unable to entirely dismantle the collegiate governance systems which by then had become par and parcel of political life. Given these conditions, it is ultimately unsurprising that so many communities ended up developing this Urban-Council system into a "Republican" style of government. I wrote several answers on this phenomenon, including this one, this other one, and this one which is purportedly about the Papal States, but has many general assertions on political life in Medieval Italy you might nonetheless be interested in.