The Italians had gained control of the Dodecanese in 1912, as part of the same war in which they gained control of Libya. Beyond the land war in Libya, the Italians engaged in significant naval warfare against the Ottomans all across the Eastern Mediterranean. As part of this they landed an amphibious force in the area and grabbed Rhodes and 11 other significant islands. This is actually where the name "Dodecanese" stems from. Rhodes was useful as a naval base during the war.

In the First Treaty of Lausanne (also called the Treaty of Ouchy) the Italians were supposed to return the islands to Turkey. But the treaty, including the return of the islands, hadn't been fully implemented when WWI broke out. After Italy joined the Entente powers Rhodes became an important base for all of them.

In 1919 a new treaty was drawn up in which the Italians would keep Rhodes while the rest of the islands would be given to the Greeks and Turks (mostly the Greeks). Italy in exchange was supposed to get some territory in Anatolia. But after the Greco-Turkish war that treaty was thrown out by the new Turkish government. In response the Italians unilaterally annexed the islands and called it a day.

So the islands weren't given to the Italians. The Italians took them before WWI and simply kept them. The reason for taking them was military usefulness. The reason for keeping them was one part, "well we're already here," and one part interest in using Rhodes as a military base, commercial hub, and transport hub.