The King James text says; “how art thou cut down,” and the italicized how shows it was placed there by the translators. The punctuation is the same in both texts. The deletions from the Book of Mormon makes the text more sensible.
Isaiah here described Lucifer’s fall from heaven in the pre-mortal existence. John the Revelator described the same event:
7 And there was war in heaven: Michael and his angels fought against the dragon; and the dragon fought and his angels,
8 And prevailed not; neither was their place found any more in heaven.
9 And the great dragon was cast out, that old serpent, called the Devil, and Satan, which deceiveth the whole world: he was cast out into the earth, and his angels were cast out with him. [Revelation 12:7–9]
Moses was told why Satan was cast out:
3 Wherefore, because that Satan rebelled against me, and sought to destroy the agency of man, which I, the Lord God, had given him, and also, that I should give unto him mine own power; by the power of mine Only Begotten, I caused that he should be cast down;
4 And he became Satan, yea, even the devil, the father of all lies, to deceive and to blind men, and to lead them captive at his will, even as many as would not hearken unto my voice. [Moses 4:3–4; see also Abraham 3:27–28]
Isaiah’s reference to Satan as a “son of the morning” (v. 12) is usually interpreted to mean that he was one of the early spirits born of our Father in Heaven. A vision of his fall, given to Joseph Smith and Sidney Rigdon, enlarges upon his objectives.
25 And this we saw also, and bear record, that an angel of God who was in authority in the presence of God, who rebelled against the Only Begotten Son whom the Father loved and who was in the bosom of the Father, was thrust down from the presence of God and the Son,
26 And was called Perdition, for the heavens wept over him—he was Lucifer, a son of the morning.
27 And we beheld, and lo, he is fallen! is fallen, even a son of the morning!
28 And while we were yet in the Spirit, the Lord commanded us that we should write the vision; for we beheld Satan, that old serpent, even the devil, who rebelled against God, and sought to take the kingdom of our God and his Christ—
29 Wherefore, he maketh war with the saints of God, and encompasseth them round about. [D&C 76:25–29]
The Saints are the members of the Church and his war with them is to oppose their following the plan of God. His objective—to become God (v. 13)—is echoed by his followers, who are collectively described as “that which … seeketh to become a law unto himself” (D&C 88:35). Lehi, getting his information from what he had read, apparently from the plates of brass, further describes Satan’s objective:
17 And I, Lehi, according to the things which I have read, must needs suppose that an angel of God, according to that which is written, had fallen from heaven; wherefore, he became a devil, having sought that which was evil before God.
18 And because he had fallen from heaven, and had become miserable forever, he sought also the misery of all mankind. [2 Nephi 2:17–18]
The common slogan “misery loves company” was probably drawn from Satan’s objective, to make us miserable because he is miserable.