Alright, I'll say it. If you don't want spoilers, don't read this.
So that appears in the no mercy route. Undyne told him that you killed lots of people and you approach to him. In defense, he fights you. Even if you do a hit with max damage or with min damage, he won't die because your screen will turn white. After a few seconds, a new screen with him and Undyne's body split in half. Monster Kid leaves and Undyne is about to die. But, she is determined too and survives. After an emotional speech, your screen will turn white and Undyne, transformed into Undyne the Undying, the second hardest boss after fat skeleton boi. Her attacks become faster, you can't flee and her HP is 23000 instead of 1500.
If you're on a run and fight Monster Kid, don't kill him unless you really want to do a no mercy run.
Hope it helps.