What does casting a spell mean?

He is correct. Casting a spell is the act of paying the mana cost in the upper-right corner and putting the spell on the Stack. Kiln Fiend's ability goes on the Stack at that poitn adn then even resolves before the spell that triggered it does. There are some abilities that also cause a spell to be cast, but they specify such in the ability (such as Chandra Ablaze's third ability)