Learn Farsi: من هنرمندم.

It is the ending م (-am) attached to the noun هنرمند (honarmand). In Farsi, present tense "to be" verbs like "am", "are", and "is" are usually attached to the end of the previous word rather than standing alone as separate words.

Do I have to say من (man) if the ending م (-am) already shows that I am talking about myself?

No, you don't. Because the ending م (-am) always means "I am", you can safely drop the pronoun من (man) and just say هنرمندم (honarmandam). Including من just adds a little bit of emphasis, like saying "I am an artist."

Why isn't there a word for "an" in the Farsi sentence?

Farsi does not use a word for "a" or "an" when stating someone's profession, nationality, or identity. You simply say the exact equivalent of "I am artist."

How do the noun هنرمند (honarmand) and the ending م (-am) connect in writing?

The word هنرمند ends with the letter د (dal). Because د is a "non-connecting" letter that never joins to the letter after it, the ending م (mim) is written right next to it without a physical connecting line. However, when spoken, they flow together smoothly as "honarmandam".