كتاب الأشربة
36  
The Book of Drinks
(18)
Chapter: It is recommended to lick one's fingers and wipe the bowl, and to eat a piece of food that is dropped after removing any dirt on it. It is disliked to wipe one's hand before licking it, because of the possibility that the blessing of the food may be in that remaining part. The Sunnah is to eat with three fingers
(18)
باب اسْتِحْبَابِ لَعْقِ الأَصَابِعِ وَالْقَصْعَةِ وَأَكْلِ اللُّقْمَةِ السَّاقِطَةِ بَعْدَ مَسْحِ مَا يُصِيبُهَا مِنْ أَذًى وَكَرَاهَةِ مَسْحِ الْيَدِ قَبْلَ لَعْقِهَا
Sahih Muslim 2034

Anas reported that when Allah's Messenger (ﷺ) ate food he licked his three fingers, and he said:

When any one of you drops a mouthful he should remove anything filthy from it and then eat it, and should not leave it for the Satan. He also commanded us that we should wipe the dish saying: You do not know in what portion of your food the blessing lies.
وَحَدَّثَنِي مُحَمَّدُ بْنُ حَاتِمٍ، وَأَبُو بَكْرِ بْنُ نَافِعٍ الْعَبْدِيُّ قَالاَ حَدَّثَنَا بَهْزٌ، حَدَّثَنَا حَمَّادُ، بْنُ سَلَمَةَ حَدَّثَنَا ثَابِتٌ، عَنْ أَنَسٍ، أَنَّ رَسُولَ اللَّهِ صلى الله عليه وسلم كَانَ إِذَا أَكَلَ طَعَامًا لَعِقَ أَصَابِعَهُ الثَّلاَثَ ‏.‏ قَالَ وَقَالَ ‏"‏ إِذَا سَقَطَتْ لُقْمَةُ أَحَدِكُمْ فَلْيُمِطْ عَنْهَا الأَذَى وَلْيَأْكُلْهَا وَلاَ يَدَعْهَا لِلشَّيْطَانِ ‏"‏ ‏.‏ وَأَمَرَنَا أَنْ نَسْلُتَ الْقَصْعَةَ قَالَ ‏"‏ فَإِنَّكُمْ لاَ تَدْرُونَ فِي أَىِّ طَعَامِكُمُ الْبَرَكَةُ ‏"‏ ‏.‏
Reference : Sahih Muslim 2034
In-book reference : Book 36, Hadith 180
USC-MSA web (English) reference : Book 23, Hadith 5049
  (deprecated numbering scheme)