Learn Farsi: دکترم و خیلی خسته‌ام. - I am a doctor and I am very tired.

Farsi allows you to drop the subject pronoun (like 'I', 'you', or 'they') because the end of the verb already tells you who is doing the action. The -am ending on both parts of the sentence means 'I am', so the word من (man) is not needed.

Why does the word for doctor (دکتر) have an -m (م) at the very end?

In Farsi, the 'to be' verbs are often attached directly to the end of nouns or adjectives as a suffix. The -m (or -am) suffix means 'I am'. So دکترم (doktoram) is a shorter, very common way to say 'I am a doctor' instead of using the full, separate verb هستم (hastam).

If both endings mean 'I am', why is دکترم spelled with just a Mim (م) but خسته‌ام has an Alef and a Mim (ام)?

It depends on the last letter of the base word. دکتر (doktor) ends in a consonant, so we just add the -am sound using the letter Mim (م). But خسته (khasteh) ends in a short 'eh' vowel sound. To prevent two vowel sounds from crashing into each other, Farsi adds an Alef (ا) as a bridge when attaching the suffix, making it خسته‌ام (khasteh-am).

Would it be wrong to add the word من (man) at the beginning of this sentence?

Not at all! You can absolutely say من دکترم (man doktoram). Leaving the pronoun out makes the sentence flow faster and sounds very natural, but keeping it is perfectly correct. You would usually only keep the pronoun if you want to emphasize that it is you, specifically, who is the doctor.