The Olympic flame may have originated in ancient Greece, where the fire was kept burning for the duration of the games, but the modern torch relay is a product of Nazi propaganda. Those Nazis sure did love their ceremonies. From The Atlantic:
Though Goebbels and Hitler both seemed to have loved the idea of the torch relay, it wasn't their idea. A man named Carl Diem, the secretary general of the organizing committee of the Berlin games, proposed it, inspired by the torch that had burned over the 1928 games in Amsterdam. Though an official in the Nazi government, Diem was a sport administrator first. After his years-long campaign to hold the Olympics in Germany had finally trickled up to the top of the government, he had lobbied, though unsuccessfully, to more freely allow German Jews to participate in the Olympics. So it's tough to blame Diem entirely for the Nazi propaganda piece that his torch relay became.So...yeah.
Whether or not Diem meant it to, a torch relay fit neatly within Nazi propaganda. Beginning the relay in Greece and ending it roughly 1,500 miles away in Berlin reinforced the idea of a shared Aryan heritage between the ancient power and the new one. It also hinted at Hitler's idea of a natural, civilizational progression from the Greek Empire to the Roman to the German. And the route happened to go through Czechoslovakia, where the stream of Nazi propaganda that surrounded it inspired some members of the ethnic German minority to clash with member of the Czech majority. Two years later, Hitler would invade and occupy part of Czechoslovakia, where he claimed the German minority was at risk.
Hitler found yet more ways to engineer the torch relay as Nazi propaganda. The head of the Reich sports office, Hans von Tschammer und Osten, convinced him to sponsor excavations of the original Olympic game sites in Olympia, further reinforcing the image of Germany as heir and caretaker of the ancient powers. Official Nazi anthem Die Fahne Hoch was played at the torch-lighting ceremony in Greece.