Learn Turkish: Onu neden partiye davet ediyorsun? O zaten evcimen. - Why are you inviting him to the party? He is a homebody anyway.

In this context, zaten is used as a discourse marker to state an inherent fact or to reinforce an argument. By saying O zaten evcimen, the speaker is pointing out a known characteristic (he is a homebody) to show that inviting him is pointless. When used this way to dismiss or reinforce a point, it translates best to 'anyway' or 'besides' in English.

I recognize ev (home) in evcimen, but what does the rest of the word do?

The word evcimen means someone who loves being at home (a 'homebody'). The suffix -cimen is quite rare in Turkish and is used to describe someone who has a strong tendency toward the root noun. Another rare example is işçimen (hardworking, someone devoted to their work). You do not need to memorize this suffix; just treating evcimen as a set vocabulary word is perfect!

Could I drop the pronoun O and just say Zaten evcimen?

Yes, you absolutely can! Because Turkish frequently drops subject pronouns, Zaten evcimen is perfectly natural. However, including O puts a little extra focus on the person, contrasting his specific personality with the highly social setting of a party.

Why use the present continuous davet ediyorsun instead of the aorist (simple present) davet edersin?