Learn Farsi: کارم را تمام کردم.

This is a "compound verb," which is very common in Farsi. It combines a regular word like تمام (complete/finished) with a helper verb like کردن (to do/make). To say "I finished" in the simple past tense, we just change the helper verb to کردم (I did). Literally, it means "I did complete."

How do we know who did the action if the word for "I" (من) isn't in the sentence?

In Farsi, the ending of the verb always tells you who did it. The ـَم at the end of کردم means "I" in the past tense. Because this ending makes the subject completely obvious, the pronoun من (I) is usually left out unless you really want to emphasize it.

Why does کارم (my work) end with the exact same sound as the verb کردم (I did)?

The suffix ـَم does double duty in Farsi! When you attach it to a verb, like کردم, it means "I" did the action. But when you attach it to a noun, like کار (work), it becomes a possessive ending that means "my" (کارم).

Why do we need the word را in this sentence?

را is a special marker that points to a specific direct object (the thing receiving the action). Because you are finishing a specific, known thing—"my work" rather than just "some work in general"—you must put را right after کارم to show it is the exact target of your verb.