This excerpt is taken from Sheikh Abu Hamza Abdul Khaliq Siddiqi's book Sahih Fazail Amaal.
عمر بن أبى سلمة قال: كنت غلاما فى حجر رسول الله صلى الله عليه وسلم وكانت يدي تطيش فى الصحفة، فقال لي رسول الله صلى الله عليه وسلم: يا غلام سم الله، وكل بيمينك، وكل مما يليك فما زالت تلك طعمتي بعد
It is narrated from Sayyiduna Umar ibn Abi Salamah that when I was a child and under the care of the Messenger of Allah ﷺ, and my hand would move around in the bowl (while eating), the Messenger of Allah ﷺ said to me: "O son! Say the name of Allah (read Bismillah), eat with your right hand, and eat from what is near to you." After that, this became my way of eating.
Reference: Sahih Bukhari, Book of Food, Chapter on Saying the Name over Food, No: 5376 – Sahih Muslim, Book of Drinks, Chapter on Etiquettes of Food and Drink, No: 5669.
Sayyida Aisha Siddiqah (RA) says that the Messenger of Allah ﷺ was eating with six of his companions when suddenly a Bedouin came and took two mouthfuls. The Messenger of Allah ﷺ said: "If he had said 'Bismillah', this food would have been sufficient for all of us. Whoever begins to eat should say 'Bismillah'."
Reference: Sunan Ibn Majah, Book of Foods, Chapter on Saying the Name over Food, No: 3264 -
Al-Albani (may Allah have mercy on him) declared it authentic.
Reference: Al-Kalim al-Tayyib, No: 112
Sayyida Aisha Siddiqah (RA) reports that the Noble Prophet ﷺ said:
إذا أكل أحدكم طعاما فليقل: بسم الله ، فإن نسي فى أوله فليقل : بسم الله فى أوله وآخره
"If anyone forgets to say 'Bismillah' at the beginning of eating, then when he remembers in the middle or after, he should say it like this: بسم الله فى أوله وآخره"
Reference: Sunan Tirmidhi, Book of Foods, Chapter on Saying the Name over Food, No: 1858 -
Al-Albani (may Allah have mercy on him) declared it authentic.
Reference: Al-Irwa, No: 1965.
The virtue of saying Bismillah and eating with the right hand can also be understood from the fact that the Messenger of Allah ﷺ said:
"When any of you begins to eat, let him eat with his right hand, and drink with his right hand, because Satan eats and drinks with his left hand."
Reference: Sahih Muslim, Book of Drinks, Chapter on Etiquettes of Food and Drink and its Rules, No: 2020.
That is, the benefit of eating and drinking with the right hand is that a person is protected from following the ways of Satan and follows the Sunnah of the Messenger of Allah ﷺ.