Learn Turkish: Lütfen kapıyı kapatma! - Please do not close the door!

Kapı means "door". The suffix -ı is the specific direct object marker (the accusative case), which shows that the action is happening to a specific door ("the door"). Because kapı ends in a vowel, we need the buffer letter -y- before adding the suffix, making it kapıyı.

How does kapatma mean "do not close"?

The dictionary form of the verb "to close" is kapatmak. If you drop the -mak, you get the basic command form kapat ("close!"). To tell someone not to do it, you add the negative suffix -ma, resulting in kapatma.

Since it says lütfen (please), is kapatma formal enough to use with strangers?

Not usually. Kapatma is the informal, singular command form used for people you would address as sen (informal "you"). Adding lütfen softens it, but if you are talking to a stranger or multiple people, you would use the formal version: kapatmayın.

Why isn't there a word for "you" in this sentence?

In Turkish, the subject is almost always understood from the verb. Commands are naturally directed at "you", so the pronoun sen (you) is completely optional and usually left out unless you want to emphasize it.