The throne is used specifically for its relationship to rule, and note in the passage in Psalms that the throne is connected to “sceptre” which is also associated with kingdom. Both the throne and the scepter are physical manifestations of the right of rulership. Isaiah sees the Lord in his capacity of the ruler of all.
The “train” is the edge of the robe of God. The image here is that no matter how high and “lifted up” this throne of God is, yet his very presence, symbolized by the hem of his garment, “fills” the temple. God is powerful, and God is present.
Textual analysis: Our current 2 Nephi 16 is the beginning of a new chapter in the 1830 edition of the Book of Mormon. This Book of Mormon “unit” of Isaiah will last until the beginning of our current chapter 23. Thus in the plates, our chapters 16-22 inclusive are given as a whole, with no chapter break in them.
Historical information: In this chapter we are introduced to Isaiah as a prophet in a time and place. Specifically, that time is “in the year King Uzziah died,” or 740/739 BC. At this time there is a divided kingdom. “The dynasty of David continued to control Judah and a fluctuating area of Benjamin. All the other tribes withdrew to form their own monarchy, which carried the comprehensive national name ”Israel“ in contrast to the retention of the tribal name ”Judah" for the southern kingdom. The two kingdoms ran parallel for 212 years, until the fall of Israel to Assyria in 722 BCE Judah survived another 136 years before succumbing to Neo-Babylonia in 586 BCE (Gottwald, Norman K. The Hebrew Bible - A Socio-Literary Introduction. Fortress Press, Philadelphia. 1985, p. 294)
“The combined work of David and Solomon moved Israel a long distance from ”chieftainship“ to ”hierarchic kingship“ along a trajectory that catapulted Israel into the forefront of ancient Near Eastern states, facilitated by the lack of any major contender for power at the time. The Solomonic bubble burst at his death when the empire split into two weaker core states after losing most of its extra-Palestinian territories. These weakened states eventually collided with the imperial designs of more powerful states. Nonetheless, the forms of state rule launched by David and maximized by Solomon set patterns that were continued in both kingdoms. It is likely that the later northern rulers Omri, Ahab, and Jeroboam II , as well as the southern rulers Uzziah and Josiah, presided for periods over empires that approached the affluence and security of Solomon’s domain.” (Gottwald, p. 323.)
Thus Isaiah comes into the picture into the southern Kingdom of Judah at a time of prosperity immediately preceding the calamity of the northern kingdom, where the 10 tribes will become “lost.”