There's definetely a bit of a gap in useful examples of written Italian which makes it difficult to confidently draw conclusions about the development of the vernacular that would become the Italian language. While there are bits here and there, and Tuscany in the late medieval period is an astounding island of vernacular literature (more on that later) it seems that every time Latin was about to outlive its usefulness, something or other conspired to bring it back. I actually wrote about that here.

But while commercial business, legal affairs, politics, and religious ceremonies were conducted in Latin well into the thirteenth century, the time when even the most erudite were speaking Latin as a first language was long gone. And while it's highly probable that the aristocrats, well-traveled merchants, and clergymen would resort to the old "speak slowly and point" technique when dealing with unfamiliar vernaculars, a very convenient phenomenon would ease communication between the well-learned: the universal popularity of Tuscan poetry and literature. Almost anyone who was in any condition to travel to a place with a different dialect would have been tutored in the tuscan "Classics," guaranteeing that someone looking to make themselves universally understood could imitate the tuscan style as best they could and would expect to be understood (at least by someone similarly well-read).

The tuscan poets were merely writing in their native language because they wanted to, and weren't conspiring to lobby for its wide adoption as Italy's standard language. Indeed, in spite of the popularity of Tuscan literature, right up to the 16th century there remained a fervent intellectual camp in favor of classical latin as the "Better Language" (Miglior Lingua) for any and all important communications. There were also people writing in eclectic and unstructured styles that merged features of late latin and central Italian vernacular. Agnolo Poliziano and his student Giovanni Pico led the Tuscan-Emilian intellectual argument in favor of this direction.

But in the sixteenth century, widely read and discussed grammatical treaties began emerging arguing in favor of the universal adoption of Tuscan as standard. The most fervent proponents weren't even Tuscan, they were Venetian: in 1516 the Venetian writer Giovanni Francesco Fortunio published Regole, a formalized grammar guide based on 14th century tuscan literature, while influential Venetian writer-philosopher Pietro Bembo published a very popular pro-tuscan treaties in 1525 titled "Prose of the Vulgar Tongue" (Prose della Vulgar Lingua). While on the surface this trend might seem surprising, it makes sense when you consider that Venice had established itself as Italy's major printing and book publishing center. Book publishers were (and still are) interested in making sure that their books are as widely read as possible, and since almost everyone buying books was tutored in the Tuscan classics it made sense for the proliferous Venetian publishers to adopt that standard, rather than lobby for a new one.

I will point out though that I'm making the process out to be much simpler than it really was: the adoption of Tuscan as the universal was far from a foregone conclusion in spite of the popularity of tuscan literature. In fact, it would seem that by the 16th century a sort of unwritten "Courtly Language" had developed as the courts of the Italian states grew closer through marriage, diplomacy, and trade. We unfortunately have close to no idea as to what this language might have sounded like or what dialect it drew most heavily from, as the keystone work by its leading sponsor Vincenzo Calmeta has been lost to time (some to refer to the Courtly Language as a "Ghost Language"). Research in the '90s (Drusi in 1995 and Giovanardi in 1998) postulates that the Courtly Language was an unstructured vernacular modeled on the language of the Papal Court in Rome, which accepted a high number of tuscan-isms but was not derivative of it. I however, have always interpreted it as having a Lombard base seeing as its two greatest sponsors, Calamera and Castiglione, were from Lombardy. Castiglione, who gives no description of the language, in his writing argues that language is a purely functional tool, and thus argued for a completely unstructured form that could even vary from court to court, as long it was intelligible.

The Tuscans themselves were occasionally reactionary with regards to the enthusiasm with which other Italians adopted their vernacular. The famous florentine statesman Niccolò Machiavelli wrote a treatise titled "Defense of the Florentine Language," where he explains to a venetian straw-man, named Messer Maffio, how illogical it is for a venetian to philosophize about tuscan grammar (as Fortunio and Bembo had done). Interest in the standardization of the tuscan language outside of tuscany might even have driven some tuscan intellectuals to develop aversion to any grammatical standardization whatsoever, arguing that the vernacular was an untamable and should remain living language (the writer Pierfrancesco Giambullari most notably lay this idea out in 1551). Others, instead, like the clique of tuscan intellectuals who came together to form the influential literary academy called l'Accademia della Crusca, published dictionaries to the effect of popularizing a very conservative take on the tuscan dialect, free from the influence of non-tuscans.

I wrote a longer and more detailed answer on the adoption of tuscan as the base of the modern Italian language here. Interestingly, what is considered "Modern Italian" is mostly the work yet another non-tuscan pontificating on the tuscan vernacular: the writer, socialite, and activist Alessandro Manzoni who headed the parliamentary commission on language standardization after Italy was unified. Manzoni's commission came up with a standardization which fundamentally consisted of Manzoni's favorite parts of contemporary florentine mixed in with his own native upper-class milanese. This means that while "Standard" Italian long aspired to imitate tuscan poets, standard Italian pronunciation and grammar is much more "artificial" than many think!