Though the book of Leviticus doesn't seem like it applies to Christians or Latter Day Saints specifically, I've actually found a lot in here that I draw a lot of strength from. I get to see the completeness of the law and the perfection of the ordinances that the Lord put forth. What has seemed like some mysterious, irrelevant, ancient, forgotten part of the law, instead is coming alive to me. I get to see the beauty that my Jewish Brethren and Sisters revere today in the same God who will reclaim them to us as well.
We have ordinances today in the temple that are different than these ones put forth in the Book of Leviticus. We don't read about the endowment ordinances in the temple though - why? Well....if they are as a sacred then as they are now - heck you weren't even to say the Lord's name anciently - then I wouldn't expect the endowment to be written in the standard works. Nevertheless we see parts of the endowment and similar truths in religions around the world - a testament to the existence of these ordinances anciently and the apostasies and scatterings that occurred anciently.
But - if you've attended the temple at all in its ordinances and how they are required to be perfect and correct - then you may be able to appreciate the completeness of the ordinances in Leviticus.
That and I *LOVE* my Savior - though I am imperfect - and I get to read a law that I know He Himself personally wrote, and then he kept.
In Leviticus I find I understand more of what He was living in His liftime, why He did some of the things He did - it just rounds out my world and testimony so much more.
We are told the Law was "dead" and hear it over and over again through the Book of Mormon and that perfection didn't come by it. Yet a close look at the ordinances therein we can see a perfection that only could come from God himself.
A close look at today's ordinances and we will see the same perfection in them both, indicating the same God wrote them both, and see a correlation between both the Mosaic ordinances and today's all sending the same message: God has atoned for us and we can be washed clean and we can become like him and we all have to submit to the same laws - including the Lord.
In Leviticus we read:
23 And Aaron shall come into the tabernacle of the congregation, and shall put off the linen garments, which he put on when he went into the holy place, and shall leave them there:
24 And he shall wash his flesh with water in the holy place, and put on his garments, and come forth, and offer his burnt offering, and the burnt offering of the people, and make an atonement for himself, and for the people. (Leviticus 16).
A few things jump out at me here:
One is that he's not wearing the ceremonial garments when he's presenting himself to make an atonement for everyone - the act of being washed is one that is a bit more humbling in an acknowledgement that we aren't worthy before the Lord so I wouldn't expect us to be wearing our finest clothes during the ordinance but rather something more humble as is described here.
Further - the priest wasn't exempt from an atonement being made for himself - it was to be offered for him and for the people. It acknowledges that he too was imperfect and in need of atonement - not a detail is missed yet here.
We know from the scriptures that the Priest or Prophet is to be to the people as the Lord is - to themhe stands in the place of God (under the Lord's direction).
So why would the priest need to be cleansed from his sins?
Well...go further....why did Christ need to be baptized? He didn't. Not to wash away sin, anyways. Yet he said that it was needed to fulfill all righteousness. And here, just as the priest in the tabernacle went to make an atonement for himself, we see the Savior also participating in the same ordinances that are expected to make us clean. The Lord was subject to the Law as well, and set up that symbolism in the tabernacle (and later, the temples) to represent this as well. We are all subject to the Father.
And as Christ said, 19 ... Verily, verily, I say unto you, The Son can do nothing of himself, but what he seeth the Father do: for what things soever he doeth, these also doeth the Son likewise. (John 5)
The father set the example for Christ, who set the example for us, who gave us ordinances to bring these things before us that we might learn to be like him - both in the temples anciently as well as today. They are the same God who gave them both, and those who kept those ancient laws are as whole as we are by keeping the modern ones - and Christ keeps and kept them as well.