The Rectification of Adam's Sin
The choicest branches of fig were singled out and used for the second arrangement on the altar, the one from which fire is taken off and brought to the golden altar, within the sanctuary. Upon this altar the incense will be burnt, and it is the incense service which was the most beloved part of the Temple service to G-d (Zohar I 130:A). Some have written (see Rashi's comments on BT Zevachim 58:A) that it was for this reason the fig branches were specifically chosen for the incense fire - for it was through the fig tree that Adam, the first man, began to make a rectification for his sin, as the Bible states (Gen. 3:7), "... and they sewed fig leaves, and made for themselves loincloths." Everything in the Holy Temple functioned on many levels; one level is the symbolic. As the incense offering was so special to the Holy One, and helped effect a righting of His relationship with man, it is fitting that such an act comes about through the very element which began that process.
After both the large arrangement and the smaller one for the incense have been lit, the priests descend from the altar and return to the Chamber of Hewn Stone for the second lottery. (Maimonides also maintains that this same priest would bring a shovelful of burning coals from the outer altar to the golden altar located within the sanctuary, upon which the incense will be burned).