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.