Learn Polish: Panie Marku, proszę zamknąć drzwi. - Mr. Marek, please close the door.

Because you are addressing him directly. In Polish, when you speak directly to someone or call out to them, you use the Vocative case (Wołacz). The word Pan changes to Panie, and Marek changes to Marku.

Why does Marek change to Marku, and where did the 'e' go?

The name Marek has a movable 'e'. When you add an ending to it, the e drops out, leaving the stem Mark-. Since masculine nouns ending in -k, -g, or -ch take the -u ending in the Vocative case, the name becomes Marku.

How does the word Pan change to Panie?

Masculine nouns that end in a hard consonant (like the n in Pan) usually take the -e ending in the Vocative case. This ending also softens the final consonant of the root, making Pan become Panie.

Why is the infinitive verb zamknąć used here?

Using the word proszę (please/I request) followed by an infinitive like zamknąć (to close) is a very common and polite way to tell someone to do something in Polish. It acts as a polite command.