"The girl asks him if he is playing with the cat."
Translation:Pigen spørger ham om han leger med katten.
Om expresses uncertainy.
Hvis expresses conditionality.
Pigen spørger ham om han leger med katten. - the girl is uncertain whether or not the boy plays with the cat.
Pigen leger hvis drengen leger (The girl plays if the boy plays) - The girl will play if (and only if) the boy is playing. Therefore her playing is conditional on the boy playing.