The aprons in the LDS Temple endowment symbolize Adam and Eve's attempt to cover themselves with fig leaves. It comes from this section of what is a VERY long ceremony - which represents the LONG ANSWER:
SCENE: In the Garden of Eden.
CONTEXT: The Temptation and Fall of Adam and Eve.
EVE: Adam, here is some of the fruit of that tree. It is delicious to the taste and very desirable.
ADAM: Eve, do you know what fruit that is?
EVE: Yes, it is the fruit of the tree of Knowledge of Good and Evil.
ADAM: I cannot partake of it. Do you not know that Father commanded us not to partake of the fruit of that tree?
EVE: Do you intend to obey all of Father's commandments?
ADAM: Yes, all of them.
EVE: Do you not remember that Father commanded us to multiply and replenish the earth? I have partaken of this fruit and by so doing shall be cast out, and you will be left a lone man in the garden of Eden.
ADAM: Eve, I see that this must be. I will partake that man may be.
(Adam takes a bite, and Lucifer walks to their side with a look of approval.)
LUCIFER: That is right.
EVE: It is better for us to pass through sorrow that we may know the good from the evil.
EVE: I know thee now. Thou art Lucifer, he who was cast out of Father's presence for rebellion.
LUCIFER: Yes, you are beginning to see already.
ADAM: What is that apron you have on?
(Lucifer draws his cloak up to reveal a black apron.)
LUCIFER: It is an emblem of my power and Priesthoods.
ADAM: I am looking for Father to come down to give us further instructions.
LUCIFER: Oh, you are looking for Father to come down, are you?
(The Gods' voices are suddenly heard in the garden.)
ELOHIM: Jehovah, we promised Adam and Eve that we would visit them and give them further instructions. Come, let us go down.
JEHOVAH: We will go down, Elohim.
ADAM: I hear their voices, they are coming.
LUCIFER: See, you are naked. Take some fig leaves and make you aprons. Father will see your nakedness. Quick! Hide!
ADAM: Come, let us hide.
NARRATOR: Brethren and sisters, put on your aprons.
(The film pauses temporarily and the lights are turned up while the patrons remove their fig-leave aprons from the bundles. All patrons tie the aprons on around their waists. The lights are again turned down and the film resumes. The camera pans down and presents the Gods' perspective as they descend. Adam hides behind foliage.)
ELOHIM: Adam ... Adam! ... Adam!! Where art thou?
ADAM: I heard thy voice and hid myself, because I was naked.
ELOHIM: Who told thee that thou wast naked? Hast thou partaken of the fruit of the Tree of Knowledge of Good and Evil, of which we commanded thee not to partake?
ADAM: The woman thou gavest me, and commanded that she should remain with me, she gave me of the fruit of the tree, and I did eat.
ELOHIM: Eve, what is this that thou hast done?
EVE: The serpent beguiled me, and I did eat.
ELOHIM: Lucifer, what hast thou been doing here?
LUCIFER: I have been doing that which has been done in other worlds.
ELOHIM: What is that?
LUCIFER: I have been giving some of the fruit of the Tree of the Knowledge of Good and Evil to them.
ELOHIM: Lucifer, because thou hast done this, thou shalt be cursed above all the beasts of the field. Upon thy belly shalt thou go, and dust thou shalt eat all the days of thy life.
AND THE CEREMONY CONTINUES . . .
Link to sources
http://www.concernedchristians.com/index.php?option=com_content&view=article&id=336&Itemid=14
http://xmo.lege.net/packham/temples.htm
And below is a photograph of what these aprons look like. As you can see they are green to mimic fig leaves.
Below you will also find a photograph of what the equivalent 2-sided Masonic apron looks like.
