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The Jews were largely exiled from 'Judea' which Romans then renamed 'Palestina' in 132CE by the Romans, after a rebellion against Roman rule. They dispersed into the Roman Empire, and gradually spread to all parts of the world. 1800 years later, there were quite a lot of them in Germany, but it's very unlikely that many, if any, had moved as individuals all the way from Judea to the territories outside the Roman Empire that are now Germany.
The Zionist movement, which started in the nineteenth century, wanted to return the Jews to their original homeland and create a new state of Israel. After the Holocaust, this project became a much higher priority, and modern Israel was founded in 1948. Jews from all over the world have settled there. However, the previous inhabitants were not happy about this.
Upvote:4
According to Max I. Dimont's "Jews, God and History" (Simon & Schuster 1962), pp. 210-211:
After the conquest of Judah by Pompey, Jews and Romans became "inseparable." Behind the Roman armies carrying the Imperial Eagles marched the Jews carrying the banners of free enterprise. The Jews were in Italy in the second century, B.C., in France in the first L> century, B.C., in Spain a hundred years later. At the end of third century, A.D., they had penetrated as far north as Cologne, Germany. When the barbarians from the East invaded Western Europe, the Jews had been there for centuries.
Of course, not all Jews went to Europe. Babylonia continued to be a hub of Jewish learning and culture, that it had been after the destruction of the First Temple, until the 11th century, A.D., corresponding with the violent invasion of the Islamic Empire. See Dimont, p. 185; Iggeret Rav Sherira Gaon ("[The] Epistle of Rav Sherira Gaon"). Also, many Jews never left Palestine, although the Romans prevented the Jewish return to Jerusalem until after the death of Hadrian in 138 A.D. Jews were again exiled from Jerusalem following the rise of the Byzantines in the third century AD. The Jewish presence in Palestine peaked in the 4th century, living in 43 Jewish communities. By 638, A.D., when the Islamic Empire took control of Palestine, at least one study estimated that between 300,000 and 400,000 Jews lived in Palestine. Jews also lived in Egypt, Turkey, Syria, Tunisia and Morocco after the collapse of the Second Temple in 70 A.D., until the establishment of the modern Israeli Jewish state.
You asked when and why Jews moved from Germany to Palestine. Emigration of Jews from Europe began at the end of the 19th century, following the Dreyfus Affair in France and the rise of Zionism, increasing following the Balfour Declaration where the United Kingdom expressed support for the establishment of a Jewish state in Palestine. Emigration from Europe to Palestine increased further with the rise of Nazism in Germany. Dimont, p. 287.
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It is not clear exactly when Jews first started settling in Germany, but there is evidence of Jewish settlement in the Rhineland communities of Speyer, Worms and Mainz already by the 10th century. There are different theories as to where they had come from.
The most widely held theory is that they were Palestinian. Most recently, this theory was challenged by Haym Soloveitchik. See his "The Third Yeshiva of Bavel" (in his Collected Essays Volume II) for a detailed description as to why he thinks that the community derived predominantly from Babylonia.
It's a technical piece, and for a more general overview, see this interview with Prof. Soloveitchik. He makes reference to this theory of his towards the end. To the best of my knowledge, it is not well accepted: the theory of Palestinian origins remains the most widely held.