Please note that this map is not north-up, nor does it seem to be using a consistent scale.

The round lake looks to be Lago Trasimeno, based on the position of the largest marked sites I can make out (Cortona, Sienna and Arezzo). The river feeding into the mystery body of water is marked as Chiana, a minor river that still flows today, and the source of the name Valdichiana ("Chiana valley").

Historically, Italy's river valleys have been plagued by marches and swamps, and this is one of them. During Da Vinci's time, there was a large, continuous swamp between what is now the town of Chuisi and the city of Arezzo. These swamps were a source of malaria and other diseases and local rulers have attempted to drain them since antiquity, with varying degrees of success.

A plan to drain the swamp we're looking at was drawn up by an Italian count named Vittorio Fossombroni in 1789, and was executed between 1826 and completed by 1838. If you can read Italian (I can't), you can find his first hand writing on this project in the 1835 work "Memorie idraulico-storiche sopra la Val-di-Chiana" (a hydraulic history of the Chiana valley). The project was a success and diverted the flow of the Chiana river into the Tiber Arno river.

Stories like this are common in Italy, with swamps being (partly) drained since the Roman republic and into the 20th century. Large swamps around the Po valley and in the Lazio area were among the last to be drained, which only happened under the Mussolini regime.

The fact that many of these lakes were not drained earlier were a combination of mostly technological limitations and sometimes the lack of political unity. Italy was fractured into many competing duchies, republics and principalities, and diverting a river often meant that a city or region could lose a source of fresh water.

Edit: please check out some corrections by AlviseFalier below.

The Chiana valley's swampiness is mentioned in passing by multiple Tuscan authors; not only by chronicles and descriptions of the landscape like Fausto degli Uberti's Dittamondo ("Quivi son volti pallidi e confusi perché l'aire e le Chiane li nemica sì che li fa idropichi e rinfusi") but also by literature like Dante's Paradiso which casually references the slow mòver della Chiana's water.

With the consolidation of Florentine control over Tuscany the draining of the Chiana became a recurring policy objective to increase the agricultural productivity and prosperity of the Tuscan hinterland. However, all attempts to do so would be unsuccessful until the second half of the 18th century, when a socialite and sometimes academic, one Count Vittorio Fossombroni, was appointed superintendent of the umpteenth drainage project after he authored a book whose title can be loosely translated as "The Swamps in the Chiana Valley and How I Would Drain Them" (edit: is the link working?). Fossombroni's plan involved re-digging the valley's river, the long-silted Clanis, in order to create a "Master Canal of the Chiana" connecting the lakes of Montepulciano and Chiusi in which to direct the valley's seasonal torrential streams.

The success of Fossombroni's plan would earn him multiple ministerial appointments within the Grand Duchy of Tuscany, and during the Napoleonic period his competences would even make him a preferred interlocutor between the Napoleonic puppet-state called the Kingdom of Tuscia and the Emperor and his staff. Fossombroni was even retained as a very prominent minister after the post-Napoleonic restoration.

The history of the Val di Chiana and of that swamp land is an interesting case study on politics and the importance of land and water from ancient times until modern days. Val di Chiana is a very flat valley that lies in-between the river valley of the Tiber to the south (which leads to Rome) and the valley of the Arno (which leads to Florence and Pisa). In the Etruscan times, and for most of history, the Chiana river in the valley would drain to the south, which is on the right in Leonardo's map. It is believed that the Etruscans first cut through the hills to the southwest of modern day Arezzo, to create a path for water to flow to the Arno.

In ancient times, more river water meant that rivers were navigable for a longer length. The plentiful and reliable waters of the Chiana valley draining into the Tiber meant easier navigation and commerce for Rome, far to the South along the Tiber. They also created the added risk of flooding, and there are many sources from the early days of the Roman empire that talk about the idea of building a dam at the southern end of the valley to mitigate that risk. Some form of dam was built under Nero, and at that time the waters in the valley were reasonably well managed and the swamp area fairly narrow, as we can tell by the fact that the Cassia, a major roman road, ran through the valley. The valley was fertile, with abundant fishing, and decent population levels.

Fast forward 1000 years of neglect, the silt brought from the many creeks around the valley into the Chiana river caused the swamp area to expand again to the area that you see in Leonardo's drawing. The darker blue lines in the middle of the water in the drawing, one leading north and one leading south, give an idea of the fact that the water from the valley was flowing towards both Tiber and Arno at the time. Monasteries in the areas had done some work in maintaining the water flows in the area, but with only limited success. Major work was hindered by the fact that the area was on the border between Florence, Siena, and the Papal lands, and no political agreement was possible.

Leonardo's 1502 trip to the Chiana valley and the map he drew had a specific purpose. Leonardo's idea was to build a canal that would make the waters navigable through the valley, and to feed some of the water from the Lake Trasimeno (on the top right in Leonardo's map), which is only 4 meters higher than the floor of the Chiana valley, into the canal to make it navigable. The idea was interesting but not feasible with the technology and funding available at the time.

With the victory of the Florentines over the Sienese in 1554, the whole valley ended up in the hands of the Medici. The southern edge of the valley, on the right in the map, was the border between the Medici's Tuscan lands and the Papal lands. That was to stay as the border between Tuscany and the Papal states until Italian unification in the 19th century. Draining the swamp in the Chiana valley meant now more fertile land for the Medici. Getting more water into the Arno meant a more steady stream, which in late medieval times was important for the wool industry that was very an essential part of the economy in Florence and in the surrounding areas. A more reliable river flow year-round meant that wool carding and other mechanized parts of the cloth production process could be done year-round and at a lower cost.

During the second half of the 16th century where was a significant amount of work done to facilitate the flow of the water out of the valley into the Arno, and some of the creek beds in the southern half of the valley were re-routed to flow towards the north side. A flood in Rome on 1598 helped to convince the pope to agree to a permanent dam on the southern end of the valley, meaning that from that point all the water would flow towards the Arno.

Some level of work in channeling the waters and draining the swamp continued through the next two centuries. The reclaimed land in the valley was fertile, as it had been in Roman times, so it was very well worth draining, but the changing political winds in the Granducato di Toscana and the availability of funds over time made it a very long term project. What you see today, basically a small canal in the middle of the valley, and plenty of agricultural land through the valley, is the result of human terraforming that has been ongoing on and off since Etruscan times.

The Web portal for the Valdichiana community has a lot more information (in Italian) on the history of the valley, and a good bibliography, for anyone who is interested in this subject.

Any study loud the ecological damage caused but swamp removal? Native species lost?