time of departure; moment of setting out
Refers to the exact moment when someone leaves or sets out. Often used in contexts like seeing someone off or timing a departure.
I noticed I had forgotten something just as I was leaving.
I saw him off at the moment of his departure.
Compound of 出 (で, 'going out') and 際 (ぎわ, 'time, occasion'). The exact historical derivation is uncertain, but the meaning is transparent from its components.