In بِيَمِينِكَ, the preposition means “with,” and يَمِينِكَ means “your right hand.” مِمَّا يَلِيكَ means “from the food near you”; يَلِيكَ conveys nearness.
In a shared dish, reaching repeatedly across others can cause discomfort. The instruction teaches respect for food and consideration for people eating together.
The general manners are clear. Special circumstances, different food arrangements, and detailed legal exceptions belong to more advanced discussions.