All perfect praise be to Allah, The Lord of the worlds. I testify that there is none worthy of worship except Allah and that Muhammad, sallallahu ‘alayhi wa sallam, is His slave and Messenger.
The hadeeth in reference is Saheeh (sound). It was cited in many Hadeeth and Sunnah books, one of which is Saheeh Al-Bukhaari, as you indicated.
The meaning of ‘Aa’ishah's words, may Allah be pleased with her, “The Prophet, sallallahu ‘alayhi wa sallam, died in my house...” is what was mentioned in other ahaadeeth; “When he (the Prophet, sallallahu ‘alayhi wa sallam) became sick, his wives said, ‘See where you would like to stay (to be nursed), and we will come to you.’ He said, ‘Do you all agree to this?’ They said, ‘Yes.’ So he moved to the house of ‘Aa’ishah, where he stayed till he died, and he was buried therein.”
“On the day of my usual turn” means that the day during which the Prophet, sallallahu ‘alayhi wa sallam, passed away was the day during which he would stay overnight in her house. (The Prophet, sallallahu ‘alayhi wa sallam, used to divide his time equally among his wives. He spent one night with each of his wives in turns, and the day when he passed away happened to be ‘Aa’ishah's regular day)
“Between my chest and my neck” means that he, sallallahu ‘alayhi wa sallam, died with his head leaning against her, between her chest and her neck.
“His saliva was mixed with my saliva” means that she used to chew his Siwaak (tooth stick) for him to soften it and give it to him (to clean his teeth) when he, sallallahu ‘alayhi wa sallam, became too weak to do that himself at the end of his life.
The merits mentioned in the hadeeth are among a great number of merits and virtues with which Allah favored the Mother of the Believers ‘Aa’ishah, may Allah be pleased with her, apart from the other pure wives of the Prophet, sallallahu ‘alayhi wa sallam, and his honorable Companions. She had every right to take pride in those merits and to speak of the blessings which Allah bestowed upon her.
Allah knows best.