We know that Adonijah was trying to usurp the throne, lining up horses and chariots ready to protect his coup (1 Kings 1:5-6). Solomon already warned Adonijah for trying to do this, telling him that he was walking on egg shells (1:52). But why was this act of taking Abishag such an egregious offense?
It could be that Adonijah was trying to usurp Solomon’s claim to being king by taking Abishag as his concubine. When David confirmed his promise to Solomon getting the throne, there was only one other eye-witness: Abishag (1:15-17). If Adonijah took her as his concubine (and he could bump Bathsheba off), it would be his word versus Solomon’s. Perhaps if he could force Abishag to confirm his story, he could take over the throne this way. He knew that this act was a way for him to take over the throne.