Brief history of Italian:

As the Roman Empire began to crumble, the languages of the decentralized Italian regions began to diverge. By the 14th century, some of these tongues were mutually unintelligible, though most were close enough. Serious writing was still mostly expected to be in Latin, but many forms of popular entertainment--love songs, low theater, graffiti--had switched to vernacular.

In around 1305, Dante Alighieri wrote a very influential treatise called De vulgari eloquentia, "on the eloquence of the vernacular." In it, he argued that Italians should stop writing in Latin and start writing in their mother tongues, because it was easier to express yourself more naturally. Here's the curious thing--Dante didn't thing of Latin and vulgar as different languages, but as different registers of the same language. This shows in the fact that he never refers to Latin as Latin, but only as grammatica, because it has rules you have to learn in school, whereas your mother tongue comes naturally. He grouped the vernaculars into three categories: lingua di oui (French), lingua d'oc (Provencal), and lingua di sì (Italian), based on their respective words for "yes." Much of De vulgari concerns the discussion of which Italian dialect is richest and most beautiful.

Dante was already a pretty famous poet when he wrote this, and The Divine Comedy later made him a legend, but he was not alone. In the same century, two other authors cemented the Tuscan dialect as literary Italian: Giovanni Boccaccio, with his collection of stories called the Decameron (which you should go read right now because it's seriously hilarious), and Francesco Petrarca, called Petrarch, another brilliant poet. The latter two were actually much more popular than Dante in Renaissance Italy, and it is their language that formed the basis of literary Italian, thanks largely to the efforts of a very influential humanist named Pietro Bembo (1470-1547). Now this language was not exactly the same as pure Tuscan dialect, but consisted of a melange of Italianized Latin and Latinized Italian. It simply leaned very heavily on Tuscan because the three great authors all happened to be of Florentine extraction. Their fame was such that people started learning their language just to read their books. Boccaccio and Petrarch became so revered for their style in prose and poetry (respectively) that you simply could not be taken seriously as an author if you did not imitate them. The first dictionaries of the Italian language contain almost no words that were not used by those two.

Thus, ironically, Latin was replaced by an equally artificial language that no one really spoke but everyone wrote in. And there's a simple economic reason for this: it's what you had to do if you wanted any chance of reaching a sizable audience. In fact, there are many examples of authors rewriting their own books to fit in. The most famous is Ludovico Ariosto, who first wrote Orlando Furioso ("The Madness of Roland," a thousand-page Arthurian epic) in his native Ferrarese, as a gift to the court of Ferrara in 1516, and then five years later produced a more Tuscanized edition for wider publication (with bonus chapters).

TL;DR I know I didn't actually say anything about Venice, but I hope that gives you some idea as to why Italians were writing in various languages. Basically, ever since the early 14th century Italian was developing as a literary lingua franca based mostly, but not entirely, on a purified form of the Tuscan dialect. It mostly existed to facilitate communication between Italians of different cities and to produce culturally unifying works of art.