God is readying the Israelites to leave Sinai. Nearly everything has been done. The tabernacle has been built, the Levites have been counted and assigned their tasks (chapter 3). Gifts have been given that will help the Levites in their work (chapter 7) and now, in chapter 8, the Levites are finally consecrated for the tasks given them.
The chapter seems a bit repetitious. Sometimes, repetition has the goal of emphasizing a point. Sometimes, the repetition emphasizes that something was done “just as the Lord commanded.” But sometimes, repetition is intended to highlight a particular point that is not repeated – and that’s what is happening here.
Notice the repetition of the following points: The ordination service for the Levites was to (A) make atonement for them (vs. 12, repeated in vs. 19) so they might (B) serve at the tent of meeting (vs. 15 repeated in verse 19). They were C) given to Aaron as helpers (vs. 16, 19) instead of (E) the firstborn of Israel (vss. 16, 18). What is not repeated is what occurs in the middle, i.e. verse 17, that this was done because God had taken them to himself at the exodus when he had killed the firstborn of Egypt. This is the highlighted point: that the Levites belong to God because of the Passover event.
There is no direct parallel to the Levites in the New Testament. The Levites are not priests, even though the priests are Levites. In the New Testament, all Christians are priests (1 Peter 2:5,9). But perhaps that simply elevates the greater deliverance Christ offered through His death over the event of the Exodus. In the Exodus, God took the Levites to be a more special people from among all God’s people. In the Christ event, God takes all His people to be more special.