The "temple garments," as they're commonly known, serve as a constant physical reminder of the covenants that the person wearing them have made. They are associated with the temple ceremonies, as you noted in the question. These ceremonies were revealed to Joseph Smith, and as they are not meant for the world in general, the details have not been published in works of scripture meant for the world in general, such as the Doctrine and Covenants.
I have not heard of any doctrine that directly equates them to the Levitical priestly garments, though Boyd K. Packer, of the Quorum of the Twelve Apostles, gave an explanation along those lines at one point, in the spirit of analogy and explanation rather than as solid doctrine:
On one occasion one of the brethren was invited to speak to the faculty and staff of the Navy Chaplains Training School in Newport, Rhode Island. The audience included a number of high-ranking naval chaplains from the Catholic, Protestant, and Jewish faiths.
In the question-and-answer period one of the chaplains asked, “Can you tell us something about the special underwear that some Mormon servicemen wear?” The implication was, “Why do you do that? Isn’t it strange? Doesn’t that present a problem?”
To the chaplain who made the inquiry he responded with a question: “Which church do you represent?” In response he named one of the Protestant churches.
He said, “In civilian life and also when conducting the meetings in the military service you wear clerical clothing, do you not?” The chaplain said that he did.
He continued: “I would suppose that that has some importance to you, that in a sense it sets you apart from the rest of your congregation. It is your uniform, as it were, of the ministry. Also, I suppose it may have a much more important place. It reminds you of who you are and what your obligations and covenants are. It is a continual reminder that you are a member of the clergy, that you regard yourself as a servant of the Lord, and that you are responsible to live in such a way as to be worthy of your ordination.”
He then told them: “You should be able to understand at least one of our reasons why Latter-day Saints have a deep spiritual commitment concerning the garment. A major difference between your churches and ours is that we do not have a professional clergy, as you do. The congregations are all presided over by local leaders. They are men called from all walks of life. Yet they are ordained to the priesthood. They hold offices in the priesthood. They are set apart to presiding positions as presidents, counselors, and leaders in various categories. The women, too, share in that responsibility and in those obligations. The man who heads our congregation on Sunday as the bishop may go to work on Monday as a postal clerk, as an office worker, a farmer, a doctor; or he may be an air force pilot or a naval officer. By our standard he is as much an ordained minister as you are by your standard. He is recognized as such by most governments. We draw something of the same benefits from this special clothing as you would draw from your clerical vestments. The difference is that we wear ours under our clothing instead of outside, for we are employed in various occupations in addition to our service in the Church. These sacred things we do not wish to parade before the world.”
He then explained that there are some deeper spiritual meanings as well, connecting the practice of wearing this garment with covenants that are made in the temple. We wouldn’t find it necessary to discuss these—not that they are secret, he repeated, but because they are sacred.
-- Preparing to Enter the Holy Temple
Likewise, the symbols on the garment are meant to serve as reminders of specific spiritual truths. Exactly what they represent cannot be effectively discussed in a setting such as this, where objectively correct answers are desired above all else, because those who have been through the temple ceremonies are under a strict covenant obligation to not reveal the knowledge gained therein to the world.
This means that anyone who claims to explain their meaning either have not actually received the ceremony and so they are just making it up, or they have received it and, in telling, are willing to break a solemn vow made to God, and thus their explanation cannot be trusted, especially as in practice, those who do are almost invariably doing so for the specific purpose of trying to make the church look bad, which should automatically call their objectivity into question.
