Probably a stupid question, but since both active voice and passive voice tranlations are correct, why not use afslutter instead of afsluttes? I think this section has indeed messed with my brain...
It's the day that's getting finished. If you say "dagen afslutter", you say that the day is finishing something. And the rest of the sentence wouldn't grammatically work either.
English has a slight problem with separating transitive and intransitive verbs, which is one of the roles of the -s verbs in Danish.