Bukhari

Hadith Commentary:
(1)
The meaning of the hadith is that whoever performed a complete ablution (wudu) that included all its sunnahs, among which is also the use of the tooth-stick (miswak).
(2)
From this, it is also understood that there is no difference between the tooth-stick (miswak) being dry or moist.
No distinction has been made between the fasting and the non-fasting person regarding the use of the tooth-stick (miswak), therefore a fasting person may use the tooth-stick (miswak).
Whether the tooth-stick (miswak) is moist or dry, whether the fast is obligatory or supererogatory (nafl), whether it is used at the beginning of the day or at its end, all of it is permissible.
Previously, Imam Ibn Sirin made an analogy between the use of the tooth-stick (miswak) and ablution (wudu).
He said:
There is no harm for the fasting person in using a fresh tooth-stick (miswak).
It was objected to him that a fresh tooth-stick (miswak) has a taste? He replied:
Water also has a taste, yet you rinse your mouth with it.
Imam Bukhari narrated the hadith of Uthman to establish the permissibility of using the tooth-stick (miswak) for the fasting person, in which the method of ablution (wudu) is described.
It is clarified therein that he rinsed his mouth and put water into his nose.
No distinction was made in this ablution (wudu) between the fasting person and the one who is not fasting.
In reality, Imam Bukhari rahimahullah wanted to refute those who consider the use of a fresh tooth-stick (miswak) as disliked (makruh) for the fasting person.
(Fath al-Bari: 202/4)

Commentary:
This hadith proves that performing khilal (passing fingers through) of the beard is a Sunnah.

That is, just as the other limbs were not washed two or three times.

Benefits:

➊ Tahawi says: This hadith is evidence that washing each limb of ablution (wudu) once is obligatory, and washing more than once—two or three times—is superior. Thus, the position of all the scholars is that this is left to the discretion of the one performing ablution: if he wishes, he may wash each limb once, twice, or three times. This concession is, in any case, established. [شرح ابن بطال: 267/1]

➋ By performing ablution in the aforementioned Sunnah manner, all previous minor sins are forgiven.

➌ After ablution, it is recommended (mustahabb) and an emphasized Sunnah (sunnah mu’akkadah) to perform two or more units (rak‘ahs) of prayer. Some of the Shafi‘iyyah hold the view that it is permissible to perform the voluntary prayers of ablution (nawafil of wudu) even during the prohibited times for prayer, because this is a prayer with a specific cause (sababi prayer). They have deduced this from the hadith of Bilal mentioned in Sahih al-Bukhari: “Whenever he performed ablution, he would offer prayer after it.” (Thus, the nawafil of wudu can be performed at any time.) And if, after ablution, one performs an obligatory prayer or specific voluntary prayers, this virtue is also attained. Similarly, by performing the greeting of the mosque (tahiyyat al-masjid) after ablution, the aforementioned virtue is also attained. [نووي: 3/ 107]

➍ «غُفِرَلَهُ مَا تَقَدَّمَ مِنْ ذَنْبِهِ» The apparent wording of the hadith includes both minor and major sins, but the scholars have specified it to minor sins, because in other narrations, with the exception of major sins, only the forgiveness of minor sins is mentioned. The forgiveness of minor sins is specific to the one who has both minor and major sins; thus, for the one who has only minor sins (then by performing the aforementioned method of ablution and prayer), the minor sins are erased. For the one who has only major sins, there is a reduction equivalent to the minor sins, and for the one who has neither minor nor major sins, his good deeds are increased accordingly.

➎ This hadith establishes the permissibility of practical instruction, because this method of teaching is more effective and makes memorization and retention easier for the learner.

➏ This hadith mentions the sequence of the limbs of ablution, because after each limb the word «ثمَ» is mentioned (which indicates sequence).

➐ This hadith encourages sincerity (ikhlas) in prayer and warns against reflecting on worldly matters during prayer, because this leads to the non-acceptance of the prayer—especially if a person has resolved to commit a sin, then in such a case, the heart is more captive to such thoughts during prayer than outside of it. [فتح الباري: 342/1]