Learn Swedish: Stäng dörren nu!

Because this is a command! To tell someone to do something, you use the imperative form of the verb. In Swedish, this form is usually the verb stem. For the infinitive stänga (to close), you just remove the final -a to get the command form: stäng.

Why does this sentence start with the verb?

Just like in English, Swedish commands start directly with the verb. You do not need to include a subject like du (you) because it is already understood from the command format.

I remember that en dörr means 'a door'. Why does it have an -en at the end here?

That -en ending makes the word definite, changing it from 'a door' to 'the door'. Since dörr is an 'en' word, we attach -en to the end of it to get dörren.

Could I add tack to the end of this to be more polite?

Yes, you can say Stäng dörren nu, tack. Swedes do not use the word tack for 'please' quite as often or as strictly as English speakers do, but adding it to a direct command like this makes it softer and more polite.