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

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.