The troll-mirror works thus: when looking in the mirror, all which is beautiful seems ugly, and all which is ugly seems beautiful.
So Kay becomes aggressive, he makes fun of Gerda, throws something after the cat, and starts taking joy in the destructive. In the end, he goes with the snow-queen to the north pole.
Gerda is deeply sad, because she loves Kay. She starts a journey, in which she goes through many sufferings, and also finds many helpers, until she finally reaches Kay in the palace of the snow-queen. His heart is frozen, and she cries for his suffering. The tears fall on his heart, and he is freed.
Thus, good is stronger than evil, and love can melt even the coldest heart, as is also the case in real life.