What is the meaning of the magi?

Strictly speaking, the Magi were a Median tribe that provided priests for the the other west Iranian tribes, including the Parsis (Persians). The west Iranians brought a primitive version of the proto-Indo-Iranian religion from their earlier homeland on the Russian steppes. At this stage, the Magi were priests of the old religion.
Zoroastrianism, derived from the same proto-Indo-Iranian religion, came later into the eastern regions and eventually spread into western Iran. The Magi continued as priests for Zoroastrianism.
Matthew was portraying the priests of this great religion as regarding the birth of a Jewish Messiah as a momentous event. By having them wish to worship him, Matthew was probably demonstrating that Jesus had dominion over even the Zoroastrians.