Learn Turkish: Ağabey ders çalışıyor.

By itself, the verb çalışmak means "to work". However, when you combine it with ders (lesson/class), it forms the phrase ders çalışmak (literally "to work a lesson"). This is the standard Turkish way to say "to study".

How do you pronounce ağabey?

Even though it is formally spelled ağabey, in everyday spoken Turkish—and increasingly in informal writing and texting—it is pronounced and spelled simply as abi (ah-bee). The soft g (ğ) and the vowels get merged together in daily speech.

Where is the word for "my"? Does this mean "My big brother"?

Yes, it usually means "My big brother" depending on the context. In Turkish, if you are talking about your own family members, you can often leave out the word for "my" (benim) and the possessive suffix. It is very similar to how we say "Mom is working" instead of "My mom is working" in English.

How is the present continuous suffix -yor attached to çalış here?