I tried to use 'that' and was marked incorrect as well. In English, you use 'that' in a restrictive way when the proceeding clause is crucial information, such as in this sentence. Which, however, is used when the proceeding clause is optional information which isn't crucial to the meaning of the sentence.
This question rejects far too many correct English translations. Please fix it or remove it from the course.
"That was the cat... " should be accepted. "...the cat that ate..." should also be accepted.
It is especially frustrating because "that was the cat that ate the fish" is almost poetic, yet is consistently marked incorrect even though it is a perfectly legitimate English.
'Which' and 'that' are not interchangeable in English! They give different meanings to the sentence. 'The cat that' implies that there is some doubt about what creature or person ate it and gives the answer - 'the cat'. 'The cat which' implies that there are several cats but this the one that ate the fish., in other words identifies the cat from amongst several.
It's so funny that most people in the comments just talk about the English rules, but we are learning Danish!
So in this sentence I think we could also use "som" --> "Det var katten som spiste fisken", but according to my grammar notes "der" is preferred in such cases (when the relative pronoun is nominative).