I would think a major factor was the prestige of the literary works from the area of Firenze. And this prestige might have reinforced the use of Toscano as a kind of lingua franca among certain people on the pensinsula way before Regno d'Italia came about. Most people however would of course only speak their local language/dialect, something that didnt really change until the advent of compulsory education.
IF literary Toscano had lacked this prestige and the political unification of the peninsula had still been driven by the Regno di Sardegna, then the standard of Italian would have been Piemontese, or if the unification had been driven by the Regno delle Due Sicilie then the base would have been some form of Napolitano,
On the other hand the literary prestige of Toscano might also be seen as one of the driving forces behind the unification as such. Without this and the notion that there was such a thing as a common 'Italian' culture then the unification might not have taken place at all!
That Toscano had a literary prestige did not necessarily imply that it was widely spoken at all.
Yes, there were other languages in the 13th and 14th century that was widely used like the Provençau of the troubadours and of course all the different regions and states had their own traditions, and also literary models. And for official purposes there was always Latin.
Rather Toscano was promoted as a language of culture by Pietro Bembo and other umanisti during the 16th century and spread by the use of printed books.
Why did Bembo actively promote the literary use of Toscano?
He grew up in Venice, traveled widely with his father the ambassador, including a stay in Firenze where he learned Toscano; he studied Greek in Messina, worked in Ferrara, was official historian of the Republic, and ended up as a Cardinal in Rome.
Because of his travels he had first-hand experience of different language in the north, center and south. I think his preference for Toscano as a common literary language was linguistic;
Toscano belonged/belongs to the central group of languages, being neither northern nor southern. He might have thought that this would make Toscano easy to understand and acquire for writers from all over the peninsula, from Turin to Palermo.