How did these wise men of the east know that a star would rise over Israel and that it would signal the birth of the Messiah around this time?

One suggestion is that the Prophet Daniel told them, or their predecessors, while Daniel was the head of the magii and the administrator of the Babylonian empire during the Jews' captivity in Babylon several hundred years earlier. Daniel could have given them the interpretation of his own prophecies predicting the 70 weeks leading up to the coming of Messiah, and he could have explained Balaam's messianic prophecy recorded in Numbers 24:14-19, "I will advertise thee what this people shall do to thy people in the latter days. And he took up his parable, and said, Balaam the son of Beor hath said, and the man whose eyes are open hath said:

He hath said, which heard the words of God, and knew the knowledge of the most High, which saw the vision of the Almighty, falling into a trance, but having his eyes open:

I shall see him, but not now: I shall behold him, but not nigh: there shall come a Star out of Jacob, and a Sceptre shall rise out of Israel, and shall smite the corners of Moab, and destroy all the children of Sheth. And Edom shall be a possession, Seir also shall be a possession for his enemies; and Israel shall do valiantly. Out of Jacob shall come he that shall have dominion, and shall destroy him that remaineth of the city."