Bukhari

Hadith Commentary:
(1)
From this hadith, it is understood that the Messenger of Allah (sallallahu alayhi wa sallam) would repeat his greetings and speech, but what is meant by this is that he would do so most of the time, i.e., he would often do this for the purpose of clarification and understanding, so that the greeting would become widespread and the matter would become firmly established in the hearts.
(‘Umdat al-Qari: 15/361)
(2)
The following explanations are possible regarding giving greetings (salam) three times:
➊ When the gathering was large and people were dispersed, he would give greetings three times so that all those present would receive the greeting. One in front, the second to the right, and the third to the left, because all those present were eager for the greeting of the Prophet (sallallahu alayhi wa sallam).
➋ If there was a huge crowd somewhere, he would give one greeting upon entering, the second upon reaching the middle, and the third upon reaching the end.
➌ This greeting is the greeting of permission (salam al-idhin), which can be said up to three times at most.
This also appears to be the intent of Imam Bukhari rahimahullah, because after this he has mentioned the incident of Abu Musa al-Ash‘ari radi Allahu anhu, which is related to the greeting of permission.
(3)
It is also possible that the first greeting was for seeking permission, the second when he approached them, and the third when he was departing from them, because in ordinary circumstances, giving greetings three times is not established from his regular practice.
The first greeting is for permission, the second for meeting, and the third for farewell.
(Sharh al-Kirmani: 2/86)
And Allah knows best.

Hadith Commentary:
It was the practice of the Messenger of Allah (sallallahu alayhi wa sallam) to give salam three times on specific occasions, for example, when seeking permission to enter someone's house. That is, once for seeking permission, the second time when approaching them, and the third time when departing from them. In ordinary circumstances, giving salam three times is not established from his regular practices. (Sharh al-Karmani: 2/86)

“Be aware, and lying”—the complete hadith is as follows: The Messenger of Allah (sallallahu alayhi wa sallam) said: “Shall I not inform you of the gravest of major sins?” We said: “Indeed!” He said: “Associating partners with Allah, and disobedience to parents.” (After this) he sat up straight, whereas previously he had been reclining, (and said:) “Be aware, and lying.” He emphasized this so much that he kept repeating it until we said: “If only he would remain silent.” (Sahih al-Bukhari, al-Shahadat, Hadith: 2654)

From this specific style of expression, it is understood that such repetition did not occur on every occasion, but rather, due to the importance of a matter, it would be repeated multiple times. Most likely for this reason, in the following narration, the restriction of (thrice) is mentioned, meaning that when he (sallallahu alayhi wa sallam) gave the necessary admonitions on the occasion of the Farewell Pilgrimage (Hajjat al-Wada‘), at the end he said three times: “Have I conveyed the message?”

Hadith Commentary:
1.
Imam Bukhari rahimahullah, with this chapter heading, has refuted those who consider the repetition of hadith or the request to repeat it as blameworthy. Repeating a statement is done in view of the differing temperaments of students; some students understand in one instance, while others require it to be repeated several times. It can also mean that if a student is slow-witted, then for him it may be repeated up to three times at most, and if he is still unable to understand after three repetitions, then there is no remedy for him. (Fath al-Bari: 1/249)

2.
One explanation for giving salam (greeting of peace) three times has already been mentioned in the previous hadith; some further explanations are as follows:
(1)
This salam is the salam of permission, which can be given up to three times at most. It is possible that this is what Imam Bukhari rahimahullah intended, because he has mentioned this hadith in the Book of Seeking Permission (Kitab al-Isti’dhan) along with the incident of Abu Musa al-Ash’ari radi Allahu anhu, which occurred with Umar radi Allahu anhu regarding the salam of permission. (Sahih al-Bukhari, al-Isti’dhan, Hadith: 6245)

(2)
When there was a large gathering and people were dispersed, salam would be given three times so that it would reach everyone: once facing forward, once to the right, and once to the left, because all those present were eager for your salam.

(3)
When you participated in a large gathering, you would give one salam upon entering, a second upon reaching the middle, and a third upon reaching the end of the gathering.

Explanation:
1:
So that the matter may be well understood.

Hadith Authentication (Takhrij al-Hadith):
[صحيح بخاري 95]

Jurisprudential Explanation of the Hadith (Fiqh al-Hadith):
➊ During speeches, preaching, and giving advice, important points should be repeated two or three times so that the audience understands and remembers them.
➋ Saying salam (greeting of peace) three times refers to giving salam when entering a house or a place. This is evident from the chapter heading in Imam Bukhari’s Book of Seeking Permission (Kitab al-Isti’dhan), and the scholars have also explained it in this way.
➌ No matter how high a person’s status or rank may be, it is necessary to seek permission before entering someone else’s house.