that game have the castle checkmate btw
HOW do you deliver checkmate with a king???
But why is it checkmate with king? If the rook is mating
Because it's wrong. The King can never deliver checkmate on its own. In order to checkmate you need to put the other King in check in a position in which it has no squares to move away from check nor block it.
Kings cannot walk next to other kings - they cannot move into check - and they only attack their immediate, adjacent squares. So Kings cannot deliver checkmate.
Discovered checkmates etc. should not count as the King is not the one delivering the checkmate.
I dont even know why such achievement exists.

But why is it checkmate with king? If the rook is mating
Because it's wrong. The King can never deliver checkmate on its own. In order to checkmate you need to put the other King in check in a position in which it has no squares to move away from check nor block it.
Kings cannot walk next to other kings - they cannot move into check - and they only attack their immediate, adjacent squares. So Kings cannot deliver checkmate.
Discovered checkmates etc. should not count as the King is not the one delivering the checkmate.
I dont even know why such achievement exists.
I think it's just a fun, unique way of checkmating. So they wanted to make it an achievement. You are 100% right, kings cannot deliver checkmate. It's always the revealing piece (or in the case of castling, the rook) that delivers the checkmate. Maybe with the wording "with" they figure as long as the king was moved or involved it counts as this unique checkmate.
They probably should have worded it a little better, like "Delivering checkmate by moving the king" or something.

...or the other way u can do it is a 0-0 or 0-0-0 (castle) ++ .
...but u can do it in a blitz game. king takes king placed next to it by accident.
Is there really a killer king achievement?
I agree with IfPatriotGames.
Does there really exist a game in which someone did a discovered attack using the king to deliver the checks of our mate?
I mean checkmate : )
i somehow did it