Andersen's simplicity of language is brilliant. He doesn't waste words in setting up the universe and scene for each of his stories. He concentrates on telling you enough and leaves the reader to fill in the gaps. This is something I think all writers should aspire to as I know I get a lot of enjoyment from working things out for myself when reading stories. I want clues. I want enough information to work things out but I don't want every little thing spelled out for me. I think good stories give the reader that room to deduce things. (This is even more crucial for detective stories but does apply to all genres).
Looking at the motivation behind the writer presenting the story as they have can be fascinating. I also love studying the motivation of the characters. I think the one question every writer should ask themselves with regard to their own characters is "is the motivation strong enough?". Also working out the structure of a story can help you stick to a good frame work for your own stories. There is no point, I feel, in reinventing the wheel when you know what structures work for what type of story.