Hadith: Whenever the Messenger of Allah (may Allah's peace and blessings be upon him) ate food, he would lick his three fingers

عن أنس بن مالك رضي الله عنه عن رسول الله صلى الله عليه وسلم أنه كان إذا أكل طعاما، لعق أصابعه الثلاث. قال: وقال: «إذا سقطت لُقمة أحدكم فليُمِطْ عنها الأذى، وليأكلها ولا يَدَعْهَا للشيطان» وأمر أن تُسْلَتَ القَصْعَةُ، قال: «فإنكم لا تدرون في أيِّ طعامكم البركة» وعن جابر بن عبد الله رضي الله عنه أن رسولَ اللهِ صلى الله عليه وسلم قالَ: "إنّ الشيطانَ يَحضُرُ أحدَكُم عندَ كلِ شيءٍ من شَأنِه، حتى يَحضُرَهُ عندَ طعامِهِ، فإذا سَقطتْ لقمةُ أحدِكُم فليَأخُذْها فَلْيُمِطْ ما كانِ بها من أذى، ثم ليَأكُلْها ولا يَدَعْها للشيطانِ، فإذا فَرَغَ فليَلْعَقْ أصابعَهُ، فإنه لا يَدري في أيِّ طَعامه البركة"
[صحيحان] - [حديث أنس رواه مسلم. حديث جابر رواه مسلم]

Anas ibn Mālik and Jābir ibn ‘Abdullāh (may Allah be pleased with both of them) reported that whenever the Messenger of Allah (may Allah's peace and blessings be upon him) ate food, he would lick his three fingers and he said: “If any of you drops a morsel of food, he should remove the dirt from it and eat it and not leave it to the devil." He ordered that the bowl should be scraped off, saying: "You do not know which part of your food contains the blessing.” The Messenger of Allah (may Allah's peace and blessings be upon him) said: “Indeed, the devil is present with you in all of your affairs; he is even present with you when you are eating. So if any of you drops a morsel of food, he should remove the dirt from it and eat it and not leave it to the devil. When he finishes, he should lick his fingers, for he does not know which part of the food contains the blessing.”
[Sahih/Authentic with its two versions] - [Muslim with its two versions]

Explanation

In the Hadīth there is a warning against the devil, and an alerting note about his permanent accompaniment of man in all of his affairs. Hence, one must guard himself against the devil, and not be fooled by that which he beautifies for him. There is a blessing in the food that man prepares, and he knows not whether this blessing is in the food he has already eaten, or in that left on his fingers, or in what is left in his dish, or in the morsel that fell. So one must preserve all of that in order to attain the blessing. "Blessing" originally refers to increase, continuity of goodness and benefiting therefrom. What is meant by “blessing” in this context is the achievement of nourishment, safety from harm, and strength in obeying Allah, the Exalted. A point of benefit that doctors mentioned in this regard is that while eating, the fingers tips release what aids in the digestion of food.