I read that here on the 5th paragraph: https://journals.openedition.org/dhfles/138
I've read some articles before showing Italian to have been the primary Western language spoken in the Ottoman empire until 16-1700, but primarily in ports among the Levantine communities, so this seems strange both geographically and temporally.
The original text of the treaty was likely Russian, as they were the victors setting the terms. While there were apparently Russian experts who knew Ottoman Turkish, there were likely no Ottoman officials able to understand Russian.
It would be unseemly for the Russians to have to make their demands in the language of the vanquished, but since the Ottomans wouldn't be able to understand Russian, a third language was needed. Italian was the best-known European language used within the Ottoman Empire, and since the Russians also had Italian speakers, they presented the Ottomans with copies of the treaty in Russian and Italian. Basically Italian was just a face-saving bridge language.
Bernard Lewis writes a bit about this in From Babel to Dragomans: Interpreting the Middle East.