Learn Arabic (Modern Standard): يا أمي، ضعي الزيت في الوعاء.

Because the speaker is giving a command to a woman (their mother). In Arabic, when you tell a single female to do something, you add the suffix ي to the end of the base command. If you were telling a man to put something, it would just be ضع without the extra letter.

Where did the first letter of the verb go? The dictionary word for "to put" is وضع.

For verbs that start with the letter و (waw), this first letter often disappears when making command forms (imperatives). So the root وضع becomes the command ضع, and for a female, we add the suffix to make it ضعي.

What does the يا at the very beginning do?

The word يا is a calling particle used in Arabic whenever you address someone directly. While we usually just say "Mom" in English to get her attention, Arabic requires you to put يا before the person's name or title when speaking to them.

How do we get "my mother" from أمي?

The word for mother is أم. The ي at the very end of the word is the possessive suffix meaning "my". So, أم (mother) plus ي (my) becomes أمي (my mother). Notice how this ي looks exactly like the one added to the command verb, but here it means "my" because it is attached to a noun!