Krishna addresses Arjuna's deepest fear head-on. Whoever thinks the soul kills, and whoever thinks the soul is killed — both are mistaken. The soul neither kills nor is killed. This statement strips away the entire foundation of Arjuna's despair. His dread was: I will be the killer of my grandsire and my teacher. Krishna responds: you cannot kill what they truly are.
Imagine watching a play. A character falls on stage, 'dies' in the story. The actor stands up after the curtain drops, unharmed. The drama was real in the moment, but the person behind the role was never in danger. The bodies on the battlefield are like characters in a cosmic play. The soul — the actor — cannot be touched by the events of the stage.
This verse appears almost word-for-word in the Kathopanishad (1.2.19). Krishna is drawing on the oldest wisdom traditions available, presenting them to Arjuna at the moment they are most needed.