Translation:HC Andersen is known for his fairytales.
That's because he was an author, not an adventurer. Although there is only one word for adventure and fairytale in Danish (eventyr), there is still a difference. If you said "Peter Freuchen er kendt for sine eventyr", everyone who has heard of him would know that you're talking about his experiences and travels, and that you're probably not saying he wrote fairytales.
While H.C. Andersen is certainly most known for his stories, he was indeed a bit of an adventurer as well. He travelled extensively in Europe and wrong large books about his travels.
in this sentence I am supposed to write fairytales, not fairy tales (in two words), so I am told that I had an extra space. There are other sentences in which it is exactly opposite, I write fairytales and am told that I had missed a space. Both forms, fairy tale and fairytale, are correct, as far as I know