The background to this is the first chapter of 1 Kings. Adonijah was the older half-brother of Solomon, but David had promised the throne to Solomon. Adonijah tried to usurp the throne by conspiring with Joab et al. to make himself king without his father’s knowing. Bathsheba complained to King David and King David put the situation right by having Solomon anointed by Zadok the priest and Nathan the prophet as the legitimate king (1 Kings 1:38–40. This is the text on which the coronation anthem composed by Handel is based: "Zadok the priest and Nathan the prophet anointed Solomon king; and all the people rejoiced and said God save the king, Long live the king, May the king live for ever," which is still sung to today—the coronation of an English monarch is a very biblical affair, with the monarch kissing the Bible and taking on oath on it to rule the nation according to the law and gospel of God etc.). Anyway, Adonijah then took himself off to get hold of the horns of the altar, which was effectively taking refuge at the altar, and asked Solomon not to put him to death for his treason (he knew Solomon was King David’s chosen heir when he tried to usurp the throne). And Solomon did not put him to death. 1 Kings 1:52 effectively states that Solomon said if Adonijah behaved himself in future we would not be punished.
This is where the text in 1 Kings 2 does not spell all the details out, and one really needs to know something of the background to the culture to understand what Adonojah was doing, but really what Adonijah then did was to try another sort of coup de’etat, more surreptitiously perhaps but none the less it was an attempt to take Solomon's throne from him, or at least the first crucial step in his plan to do so. In those days polygamy was practised, and kings (and others) had many wives and also concubines. In his old age David was cold in bed and so they brought Abishag to him to keep him warm in bed (I have no idea why one of his wives could not have kept him warm in bed!). The Bible says that Abishag did not have sexual relations with him, and so she was not really a concubine, but just ministered to his needs, more like a cross between a nurse and a hot water bottle. But she was probably perceived by the outside world as a concubine, part of King David's harem. And this is the point.
Adonijah claimed the throne was his by right of primogeniture, even though it had been given to Solomon, so he still believed he was the rightful king. This is why Bathsheba asked if he came in peace—she knew he believed he had a rightful claim. She naïvely believed him when he said he had come in peace. But then he asked something that Bathsheba should have known was a deceit, i.e. marriage with Abishag. If there was no problem with the marriage why did Adonijah not ask Solomon himself? He asked Bathsheba to ask for him because he was playing a high stakes game of political intrigue and he thought that by using Bathsheba he would perhaps be able to outwit Solomon and keep himself out of danger; perhaps he thought that Solomon would be weak enough to grant his request to his mother, when he knew he would not grant it directly to him—he was acting deceitfully. Solomon realised what had happened, i.e. that he was trying to get the kingship by devious means. This is what Kiel and Delitzsch have to say about Adonijah’s plan: "Although Abishag had been only David's nurse, in the eyes of the people she passed as his concubine; and among the Israelites, just as with the ancient Persians, taking possession of the harem of a deceased king was equivalent to establishment of the claim to the throne (see at 2 Sam. xii: 8 and 2 Sam. iii:7, 8)" (Kiel and Delitzsch, Commentary on 1 Kings, p. 32). 2 Sam. 3:7, 8 and 2 Sam. 12:8 confirm this practice, as does 2 Sam. 16:15–23, in which Absalom, having tried to usurp the throne of David, goes into his concubines.
For Adonijah to marry Abishag was a way of claiming his right to the throne, and Solomon would have perceived, I think correctly, that he was up to no good and was intriguing to overthrow Solomon by talking possession of David's concubine and thereby trying to make himself look like the rightful heir.
Well, there you have it. Politics and politicians—what a lot! The symbols change but not the intrigue. Abishag was a symbol that Adonijah needed for his claim to the throne. This is not a romantic story about two lovers; it is about brutal, nasty politics, in which innocent people (Abishag and Bathsheba in this case) are used by those playing power politics.
Answered by Stephen Perks