Bukhari

Hadith Commentary:
Here, "rumal" refers to a cloth that is used to remove grease from the hands after eating. The Prophet (sallallahu alayhi wa sallam) instructed to clean the hands by licking the fingers and then wiping them with this cloth. Although the word "rumal" is not explicitly mentioned in the hadith, the Imam has alluded to another narration of the hadith, which Muslim has recorded. Its wording is: "fa la yamsahu yadahu bil-mindeel," meaning that one should lick and clean the hands before wiping them with the cloth.

Hadith Commentary:
(1)
Although this hadith does not explicitly mention licking or sucking the fingers, nor does it mention a handkerchief, however, in a narration from Jabir radi Allahu anhu, it is stated that until he licks his fingers, he should not wipe his hand with a handkerchief; he does not know in which part of the food the blessing has been placed.
(Sahih Muslim, Al-Ashribah, Hadith: 5300 (2033))
In Musannaf Ibn Abi Shaybah, it is mentioned that when one of you finishes eating, he should suck his fingers before cleaning his hands.
(Al-Musannaf li Ibn Abi Shaybah, Chapter on Licking the Fingers, No: 24437)
(2)
From this hadith, it is understood that after finishing the meal, it is not permissible to use a handkerchief or tissue paper until the fingers have been licked, lest the food stuck to the fingers be wasted and the blessing of the food be lost.
We should strive to revive this Sunnah in general gatherings and should not feel disdain or aversion in cleaning the fingers by licking them.

Hadith Commentary: Benefits and Issues: When a person eats food, some portion of it sticks to the person’s hands, fingers, and the vessel, and the person is unaware in which part of the food the blessing lies. Therefore, the Prophet (sallallahu alayhi wa sallam) encouraged licking the hands and the vessel and cleaning them thoroughly before washing and cleaning them. The way the fingers are cleaned well by the saliva of the mouth, such cleanliness cannot be achieved with tissue paper. Furthermore, this also proves that even the smallest portion of food should not be wasted. However, today, in imitation of others, instead of acting upon this golden principle of Islam, we use tissue papers as a fashion and consider cleaning the fingers and the vessel as uncivilized and a sign of greed or miserliness. One may also let his beloved wife or children lick his fingers, because they will not dislike it, and they will also come to know of this principle. According to some, here “or” is for variation, i.e., you may do this or not, but rather it is for doubt, meaning whether the Prophet (sallallahu alayhi wa sallam) said this word or that word. In this case, the meaning will be: let his mouth be licked. The purpose of both words will be the same, that he should lick it himself.

Benefits and Issues:
➊
This was a reflection of the refined nature of the Messenger of Allah (sallallahu alayhi wa sallam), that while eating, he (sallallahu alayhi wa sallam) would use three fingers—that is, the thumb and two fingers—instead of all five fingers.
(See: Hadith 3848) Food is meant to be eaten, and whatever remains on the fingers, there is no justification for wasting it. The most appropriate thing is to eat it. The bonds of love and closeness that exist within a family are such that family members eat each other's leftover food, which is a very endearing and beloved act. Furthermore, if while eating, some gravy gets on the fingers, and one lets his children, wife, or other family members lick it, then this act is permissible and allowed.

➋
After licking or letting someone else lick the fingers, it is permissible to clean the hands with a cloth. And it is not a legal requirement in Shari‘ah to immediately wash the hands with water; however, it is better to wash them before going to sleep.
See: Hadith 3852

Takhrij:
[بخاري : 5456],
[مسلم، الاشربة 5294],
[بلوغ المرام : 1241],
[تحفة الاشراف 88/5 و 945]

Benefits:

➊ By “cleaning the hand” in «لا يمسح يده», what is meant is cleaning the hand with a handkerchief or towel, as is narrated in Sahih Muslim from Jabir radi Allahu anhu that the Messenger of Allah sallallahu alayhi wa sallam said:
«ولا يمسح يده بالمنديل حتى يلعق أصابعة»
“Do not clean your hand with a towel until you have licked your fingers.” [الأشربة 134]

In the early days, the Companions did not have towels, and food prepared with fire was also rare. In those days, after licking their fingers, they would clean them on their palms, wrists, or feet. And they would perform prayer without (repeating) ablution (wudu). [بخاري عن جابر 5457]

➋ The reason for licking the hand has been explained by the Messenger of Allah sallallahu alayhi wa sallam himself.
«فائه لا يدري فى أى طعامه تكون البركة» [مسلم عن جابر /الأشربة 35]
That the one who eats does not know in which part of his food the blessing (barakah) lies.

➌ For this reason, the Messenger of Allah sallallahu alayhi wa sallam commanded that the bowl be cleaned, and also said that if a morsel falls, he should pick it up, clean it, and eat it, and not leave it for Shaytan. [مسلم الاشربة : 137]

➍ The meaning of barakah (blessing) is that it is easily digested, fully assimilated into the body, does not become a cause of illness, and becomes a means of assistance in obedience to Allah. «والله اعلم» [نووي]

Barakah also includes that the feeling of hunger is removed by it, because sometimes a person eats a lot of food, his stomach is filled, but his hunger is not satisfied, his greed does not end, rather he keeps eating, and eventually that food becomes a burden and a cause of illness for him. And sometimes, after just a few morsels, one feels satiated and experiences the best sense of refreshment—this is the effect of the barakah that he attained in some morsel of the food.

➎ From this hadith, it is also understood that washing the hands after eating is not necessary; merely wiping them is sufficient. See also benefit number 1 of this hadith.

➏ “He should lick his hand himself or have someone else lick it.”
That is, someone who does not feel repulsion at licking his hand, such as a wife, child, or brother, etc. If he has a goat or cow lick it, that is also permissible, as the blessing is not wasted.

Takhrij:
«أخرجه البخاري، الأطعمة، باب لعق الأصبابع ومصها قبل أن تمسح بالمنديل، حديث:5456، ومسلم، الأشربة، باب استحباب لعق الأصابع والقصعة...، حديث:2031.»©Explanation:
➊ In this hadith, attention is drawn to one of the etiquettes of eating, which is that after eating food, before cleaning the hands with a napkin or the like, one should clean the fingers by licking them oneself or having someone else lick them.
It is quite possible that the blessing (barakah) lies in the very food that remains on the hand.
➋ During the meal, it is not correct to keep cleaning the hands with a napkin or the like, or to clean them before licking the fingers.

Benefit:
From this hadith, it is understood that after eating, one should not immediately wash the hands nor wipe them; rather, one should lick them. Allah Ta’ala knows best in which morsel of food He has placed the most blessing. The explicit mention of blessing is found in Sahih Muslim: 5300 (2033): After licking the fingers, one should wash them or clean them with a handkerchief or tissue paper.