Matthew does not identify the magi, but we know they come from the East. If one is in Bethlehem, to the east probably means somewhere in Mesopotamia (present-day Iraq). Scholars seem to be pretty much unanimous that these men were from Chaldea/Babylon. The Chaldeans were well know for their interest in astronomy from ancient times. The magi are traditionally known as “wise men” in the Roman Catholic and Orthodox tradition. I suppose that the men were wise. After all, they outwitted Herod the Great (Matthew 2:16), but the word magi does not necessarily mean wise. It can be translated as sorcerer or priest. Let us say that it is very likely that the magi were priests of a Near Eastern religion, probably from Babylon, who were acquainted with the God of the Jews.
Why are they important? They are important because they announced to the world the coming of the Messiah! It is significant in Matthew that the ones who proclaimed Jesus to be king and Messiah were Gentiles, not Jews. I would say that what is most significant about the magi is that they publicly declared Jesus to be the king, both of the Jews and Gentiles.
What would the Jews have thought about them? We need to be aware that there was a very large population of Jews in Mesopotamia at the time. The largest colony of Jews outside of Canaan was almost certainly in present-day Iraq in the first century. In fact, a large portion of the Jewish Talmud, scholarly work by Jewish teachers in the first and second century is known as the Babylonian Talmud. Therefore, Jews in the time of Jesus would have been aware that these magi were very familiar with Jewish teaching and customs. The magi were certainly aware to some extent of the significance of the birth of a baby who would be king in Israel. It is likely that they were aware of the prophecies of the Jewish Messiah. It might be a little like if a Jewish person were to interact with a Christian group. Christians would assume that a Jew understands quite a bit about Christianity. It is somewhat likely that the magi were monotheists with sympathy for Judaism.