Learn Turkish: Ailesini çok özlüyor. - He misses his family a lot.

Because "his family" is the specific, known object of the verb özlüyor (misses). This means it takes the accusative case ending. If you were to ask "Whom does he miss?" (Kimi özlüyor?), the answer is Ailesini. Notice how the specific object gets the same -i ending as the question word kimi!

What is the exact breakdown of all the suffixes on Ailesini?

It breaks down like this: aile (family) + -si (his) + -n- (buffer letter) + -i (accusative case). Whenever you add a case ending (like the accusative -i) after a third-person possessive ending (like -si), Turkish requires you to use the buffer letter -n- in between them.

The dictionary says the verb "to miss" is özlemek. Why does it become özlüyor instead of özleyor?

This is due to a vowel change rule! When you add the present continuous suffix -yor to a verb root that ends in -a or -e (like özle-), that final vowel gets "narrowed." Because of vowel harmony rules, the e in özle- turns into an ü before -yor, giving us özlüyor.

Where is the word for "he" in this sentence?