Thank you for the question on the Magi, “Why do the 3 Magi show up in so many manger scenes?”
The primary reason for three Magi coming to see Jesus at a home is because of the three gifts that were given to him: gold, frankincense, and myrrh. Each magi presenting one gift to the newly born baby.
Matthew 2 says this “After they had heard the king, they went on their way, and the star they had seen when it rose went ahead of them until it stopped over the place where the child was. 10 When they saw the star, they were overjoyed. 11 On coming to the house, they saw the child with his mother Mary, and they bowed down and worshiped him. Then they opened their treasures and presented him with gifts of gold, frankincense and myrrh. 12 And having been warned in a dream not to go back to Herod, they returned to their country by another route” (NIV).
The text uses the plural pronoun, they, which means that there were at least two magi. With the three presents given to the newly born infant, it is presumed that there were three magi.
As one thinks more on the text, it is more than likely that a caravan traveled with them for they needed the supplies for themselves and their animals. The travel time would have taken several months since they would travel at night following that star. It took about 4 months for Ezra to reach Jerusalem and that was traveling by day. The servants who traveled with the Magi are not mentioned at all.
One additional note is that the Magi did not come to the cave but to a house. Verse 11 says this, “11 On coming to the house, they saw the child with his mother Mary, and they bowed down and worshiped him.” From a production viewpoint of the birth of Jesus, some liberty may be given to the writer in combining the scenes together into one. The Biblical narrative states that by the time the magi arrived, Joseph and Mary lived in a home.
SUMMARY: The magi and their servants were present at the home of Joseph and Mary. The three gifts that were presented to Jesus suggests that three magi entered the home of Joseph and Mary.
For more perspectives:
https://www.quora.com/Why-do-the-3-Magi-show-up-in-so-many-manger-scenes