u/PhiloSpo gave a good answer regarding the origins of the "three kings" and their role in Matthew's narrative. I'm going to focus more on how a 1st-3rd century audience in the Roman world would probably have understood the magi, hearing the story for the first time. I've written about similar ideas in the whole span of ancient Greece and Rome before: here for Zoroastrianism broadly, and here for the Magi specifically.

The author of Matthew pretty clearly seems to imply the Magi in the sense of followers of Zoroastrianism through their use of astronomy/astrology. Ironically, this isn't actually accurate to actual Zoroastrian practice. There was no particular emphasis on astronomy among the actual Zoroastrians, but during the Hellenistic period, the misconception became widespread. Zoroaster himself was misunderstood as a Babylonian (hence the "Chaldean" ending on the Greek name Zoroastres), and conflated with Babylonian astronomy. He was actually credited by some authors with inventing the practice. As a result, his followers, correctly identified (in part) as the magi were also associated with astronomy.

The Greek world absolutely did have some fundamental misunderstandings of the Magi, including their association with magic, as it is in fact the root word of "magic." By the Roman period, it was sometimes used euphemistically, but "Magi from the east who watch the stars by night" would still probably have been a clear enough reference to other established knowledge of the magi to make it clear these weren't random magic users. Matthew's description includes enough other details to emphasize their role as Zoroastrian priests to his audience.

Strabo, and Plutarch both provide descriptions that imply that the Roman world had finally come to understand the basics of what their eastern contemporaries believed after a few centuries of regular contact. Of course, the Greeks and Romans hadn't been completely ignorant up to this point, but there were many misconceptions, which did continue to circulate and evolve into the medieval period. One of the earlier examples of a Greek writer understanding Zoroastrian beliefs is the Oracles of Hystaspes. This is an apocalyptic text, written at some point during the Hellenistic period, but attributed to Hystaspes, the first king to convert and shelter Zoroaster (Vishtaspa in the original Avestan). In a way, it's very similar to a Zoroastrian book of Daniel.

The Oracles are mostly lost to us today, but seem to be a fairly accurate assessment of the end of days as described in Zoroastrian literature, called Frashokereti. We know it was discussed and heavily quoted by early Christian authors Clement of Alexandria, Justin Martyr, and Lactantius, who all actually accepted it's prophetical nature and interpreted it as describing the fall of Rome and the return of the Messiah. So by the second century we can already see examples of Christians identifying with Zoroastrian messianic beliefs. This may give us a hint to how they understood the Magi, as we know the magi were still associated with Zoroaster at that point. One valid interpretation is that the Magi came and acknowledge Jesus as the fulfilment of their own religious prophecy. This would also have been fuel for the pro-gentile camp when early-proto-Christians were still debating whether or not to proselytize non-Jews.

Additional reading suggestions:

Traditions of the Magi: Zoroastrianism in Greek and Latin Literature by Alfred de Jong

A History of Zoroastrianism, vol. 3 by Mary Boyce and Frantz Grenet

"The Zoroastrian Doctrine of Salvation in the Roman World: A Study of the Oracles of Hystaspes" by John Hinnel

and less directly related, but I always recommend Zoroastrianism: An Introduction by Jenny Rose as something to correct some of Mary Boyce's more outdated theories, though I don't think any of them play into the issues discussed here.

Overall lack of textual sources make this probably impossible to answer historically, nor what were the intentions of the author. Perhaps this brief account will be rather perplexing ( so I will link some of the relevant literature for further ) and it probably will now answer the question straight on ( I do not see how anyone could do it. ). I should say that this is not a settled matter on numerous accounts, so perhaps some might approach it differently.

1. Magi as Royalty and other possible connections
2. Contemporary reader-response.
3. Some sources and further readings for Matthean Nativity narrative

Magi as Royalty

We can attest that the identification of Magi as Kings can be attested quite early, from the texts like Cave of Treasures, traditionally attributed to Ephrem the Syrian ( 4th AD ), though modern scholarship has seriously disputed the authorship, and is thought to be from 6th AD. Another quite known reference is made in Sermo 139 by Caesarius of Arles. Augustine preaching to Epiphany writes:

King Herod, you see, was afraid, when the Magi told him about the child... How dread must be his judgement seat, when his infant´s cradle could so terrify the pride of kings! How much more prudently do kings nowadays, not seek like Herod to kill, but rather delight like the Magi to worship him.

so we can be quite certain that by sixth century, the picture of Magi as Royalty is quite widespread and common ( and was undisputed until Reformation ).

Some of it can be found in Old Testament:

The Gentiles shall come to your light,And kings to the brightness of your rising ( Isa 60:3 )

The kings of Tarshish and of the isles Will bring presents;The kings of Sheba and Seba Will offer gifts. Yes, all kings shall fall down before Him; All nations shall serve Him. ( Psalm 72:10-11 )

Perhaps other connection that can be found through Origen in Homilies in Numeri is that Balaam was a founder of the order of the magi, so arguably the magi are his successor, and in Numbers 24, Balaam´s final Oracle we find:

“I see Him, but not now; I behold Him, but not near; A Star shall come out of Jacob; A Scepter shall rise out of Israel, And batter the brow of Moab, And destroy all the sons of tumult. ( Numbers 24:11 )

So we get a somewhat fulfilling prophesy in the Magi following a star said to be a sign of a Messiah as prophesied by Balaam.

Contemporary reader-response, if we put the above said in opposition with what we can reasonably assume would be common knowledge at the time of Matthew, and we can contrast this with a broadly Greco-Roman discourse of Magi. For instance, in Strabo ( Geography 15.1.68 ), we find "[Magi] attend the Persian Kings, ... guiding them with their relations with the gods.", or were educators of royal children.

Two two other famous stories, Tiridates and Gaumata, the former attested by writers such as Pliny, Dio Cassius, the latter by Herodotus, Justin, Strabo, Josephus, and so forth. Both of these stories rather dissociate the magi and the King, and perhaps imply that being one disqualifies the second. After the death of Gaumata ( per Herodotus History 3.79 ) we see the slaughter of the Magi, and we see them ( or rather, they would see them ) as an object of royal oppression and persecution. So it would be highly unlikely for Matthean contemporary to identify Magi as Royalty, since the known discourse would put them in opposition.

And this conclusion is likewise in line with Jewish Midrash and Haggadah, where we get Balaam ( and the magi ) is a servant and lacks any kind of political power, and is summoned by a king.

Then the king gave the command to call the magicians, the astrologers, the sorcerers, and the Chaldeans to tell the king his dreams. So they came and stood before the king.

“My decision is firm: if you do not make known the dream to me, and its interpretation, you shall be cut in pieces, and your houses shall be made an ash heap.

They answered again and said, “Let the king tell his servants the dream, and we will give its interpretation.”

( Daniel 2:2, 5, 7 )

Similar attitude can be found from Herod in Matthew.

And the overall Matthean narrative might push for taking Magi as servants by contemporaries, as throughout Matthew we find that worldly power is consistently aligned with Satan and worldly rule.

But many who are first will be last, and the last first. 19:30

And God is associated with those who lack power, the servants, the meek, children, and so forth ( 10:24, 20:27, 5:5, 18:1-4, 19:13-14, 11:25, 18:6 ), so before a more dominant more secular Augustinian readings by 4th century, which could make the transition, we can reasonably assume the contemporary reader would view magi as not Kings. Mind, Augustine had predecessors and a gradual shift to the position of Magi as Kings.

As always, open to subsequent and requested clarifications, as I am afraid I have not answered satisfactory or what was hoped for, probably. I will link some relevant literature to the narratology itself, and how Matthean narrative of nativity was probably synthesized from two different sources.

Literature:

Powell, M. A.: The Magi as Kings: An Adventure in Reader-Response Criticism. The Catholic Biblical Quarterly Vol. 62, No. 3

Nolland, J: The Sources for Matthew 2:1-12. The Catholic Biblical Quarterly Vol. 60, No. 2

Raymond E. Brown: The Birth of the Messiah A Commentary on the Infancy Narratives in the Gospels of Matthew and Luke ( 1993 )

Trexler, R.C: The Journey of the Magi ( 1997 )

Luz U.: Matthew 1-7 A Commentary ( 2007 )

EDIT: thanks to /u/Trevor_Culley, spotting some poor formatting/poor writing on my part that was bound to lead to misinterpretation.