Our story begins with the Italo-Turkish War of 1911. The war broke out when Italians, after long years of negotiations with other European powers and plans, decided to invade and colonise Libya. Libya then was formally part of the Ottoman Empire. However, with the capture of Egypt by the British in 1882, it no longer had any land connection to the rest of the empire. With the Ottoman Navy not being in excellent condition for a long time and especially suffering decay under Abdulhamid II, the naval connection was, at best, shaky as well. It was in this context that the Italians started their military operations in Libya.

Contrary to what all this might suggest, and despite some revolutionary military tactics such as the very first example of aerial bombardment, the Italians did not occupy the whole of Libya quickly. The local forces of the Ottomans, supported by local tariqahs and tribes, would prove to be formidable opponents against them in guerilla warfare, especially far away from the coast. The Ottoman government also managed to support its forces on the ground and the tribes by sending a group of young officers, including Enver (later became a Pasha and the Minister of War during the WW1) and Mustafa Kemal (later became Atatürk) Beys. As a response, Italians decided to hit Ottomans elsewhere, using the advantage of their navy. A particular place they hit the Ottomans were precisely those islands in the Aegean Sea called the Dodecanese Islands. Italian Navy surrounded them, landed troops, and soon the islands were captured one after the other by the Italians. The Italian government intended this as leverage against Ottomans, to force them to come to terms with the Italians, to leave Libya in exchange of the islands. The plan worked as the Ottoman government signed the Treaty of Ouchy in 1912 with Italy, according to which Ottomans would leave Libya but take back the Dodecanese Islands in return.

Yet, the Ottomans had enemies elsewhere with plans of their own. Before the Treaty of Ouchy could be implemented, the First Balkan War broke out. The Balkanian states of Bulgaria, Greece, Serbia, and Montenegro attacked the Ottoman Empire and recorded noteworthy victories. Especially the Bulgarians proved particularly effective and, with the internal political divisions within the Ottoman Army, they took Edirne and came very close to the Ottoman capital. The First Balkan War was followed very soon by the Second Balkan War, which ended in 1913 and saw Edirne return to the Ottoman Empire. Around a year later, the First World War would break out. As you can see, the Ottomans did not particularly have any time to do much about the fate of the islands at all, which remained under Italian hands as a result.

From the WW1 the Ottomans emerged defeated and forced to sign the Treaty of Sevres in 1920. The relationships between Italy and its allies at this time were quite strained, in part due to their ideas of sharing Anatolia being worlds apart (though they also had disagreements elsewhere, in particular on the Italian-Yugoslavian border). Italians thus obtained significant portions of Southern Anatolia (including Antalya and Konya) but agreed to hand over the Dodecanese Islands to the Greeks in exchange. Yet, this treaty too failed to materialise because of the Turkish War of Independence (1919-1923). With the Turks defeating the Allies and nullifying the Treaty of Sevres, the status of the islands became once more up for debate/negotiation. As a result of the Turkish War of Independence, the Treaty of Lausanne was signed, with which Turkey recognised Italian rule in the islands. Thus they de jure became part of Italy too, following a period of over 10 years under de facto Italian control.

Until the WW2, the islands remained part of Italy. With the surrender of Italy in 1943, the islands (which were used by Italians as naval bases for their campaigns elsewhere) became a battleground between the Allies (including Greece) and the Axis (by now just Germany). With the Germans losing, the legal status of the islands once more became subject of diplomatic debate. As the Greeks were part of the winning side and as the islands had Greek majority, it was deemed appropriate for them to become part of Greece. At this time, Turkey did not have much say. It did not exert any actual control over the islands for the last 30 years and formally recognised the loss of control in 1923. The islands' Turkish population was by now very much limited. Turkey was also not formally part of the war until its very end. The only way Turkey could reestablish control over the islands was joining the War on the opposite side of Greece, which would mean an Axis Turkey, which in turn would probably mean a Soviet-occupied Turkey at the end of the War. With hindsight, it seems not joining the War for the islands was the more prudent decision... That said, I know that some circles in Turkey are still quite shocked by how Turkey does not control those islands so close to it. That belongs more to modern Turkish politics than history though, so I will leave it here.

TL;DR: Italy took them from Turkey during the Italo-Turkish War of 1911-12, Turkey did not (and could not) interfere later, Italy lost the WW2, and the islands became Greek.