simple: you both did damnage to each other, then you hit him with that demoman melee and you took explosive damage + fall damage, then died after you killed him. Hence him "finishing you off."
Hopefully I explained this well enough.
I believe he's referring to the self-assisting bug. Basically valve decided that you and your spirit are two different people, so when you are alive you do damage, and when you're dead you're a separate entity doing damage. They claimed to have fixed that in patch notes though.