We may find one explanation that is based on lineage. This is outlined first. The second explanation is based on a tithe (tenth). The second explanation may be found in the Book of Jubilees. This is shown second.
Moses and Aaron
The original reason for having a separate priestly tribe is because the children of Israel were afraid of God and His voice. They wanted someone to stand in their place. This fact appears to be the only reason why the Levitical tribe was chosen to represent the other children of Israel.
And all the people saw the thunderings, and the lightnings, and the noise of the trumpet, and the mountain smoking: and when the people saw it, they removed, and stood afar off.And they said unto Moses, Speak thou with us, and we will hear: but let not God speak with us, lest we die.
So, they chose Moses the son of Amram the son of Kohath the son of Levi (see Exo 2:1-2, 6:18, Numbers 4:4-14).
Moses had led them out of Egypt with the help of his brother Aaron. These two were Levites (see Exo 4:14).
The assumption was that because Moses could behold God and live that his brother Aaron and his sons would be able to do so also.
Subsequently, while Moses is on the Mount receiving the 10 commandments, Aaron and the others are in the valley below collecting gold and building a golden calf.
Moses descends, sees their idolatry, and cries out, whoever is for the LORD, rally to me. The sons of Levi is the only tribe to come to his side.
This is one reason the Levites were chosen to be the only priests under the Mosaic Covenant.
And when the people saw that Moses delayed to come down out of the mount, the people gathered themselves together unto Aaron, and said unto him, Up, make us gods, which shall go before us; for as for this Moses, the man that brought us up out of the land of Egypt, we wot not what is become of him.
Then Moses stood in the gate of the camp, and said, Who is on the LORD'S side? let him come unto me. And all the sons of Levi gathered themselves together unto him.
In short, Moses was a Levite who was able to "withstand" God. Aaron was his brother. Moses anoints Aaron into the priesthood.
This was an influential book that has an oral tradition. Some groups believe it to be scripture. It gives three ideas, but I focus only on the third one as the most intriguing and sensible explanation.
Jacob had 12 sons. Reuben, Simeon, Levi, Judah, Isaachar, Zebulon, Joseph, Benjamin, Dan, Naphtali, Gad, Asher.
Jacob had promised to tithe everything to God after his dream. This would include his sons.
And this stone, which I have set for a pillar, shall be God's house: and of all that thou shalt give me I will surely give the tenth unto thee.
So, the explanation goes, if Jacob counts backward from his youngest son, the 10th son is Levi. Thus, Levi is tithed to God in the priestly service.