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 I did at one point hear a somewhat similar explanation from a top church leader. (Vague memory, and I have no reference to link to, but I remember that this was given in the spirit of analogy and explanation rather than as solid doctrine, during a discussion involving leaders of other churches.) It went something like this:
In your church, you have people who hold clerical offices. This
carries certain rights and responsibilities with it. And they wear
special clerical clothing as a symbol of their office. Well, for us it's a
very similar system, except that we have a lay priesthood that any
worthy member can be ordained to, not just leaders. These people have
their own vocations and day-to-day activities, and it would be
impractical for them to go around dressed like priests all the time,
but the temple garments allow them to carry such a reminder of their
ordination with them at all times as a personal reminder, to them, of
what it means, instead of an outward symbol to the rest of the world.
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.