Learn Hardcore Korean: bihaenggiga gonghangeul tteonayo.

-가 is the subject-marking particle attached to a noun ending in a vowel. Here 비행기 (airplane) becomes 비행기가, marking “the airplane” as the subject of 떠나요 (leaves/departs).

Why is 공항 followed by -을 instead of -에 or -에서?

In this sentence, 떠나다 is used transitively, so the place being left takes the object marker -을. If you treated 떠나다 intransitively you could say 공항에서 떠나요 (“leaves from the airport”) using -에서 to mark location. But 공항을 떠나요 (“leaves the airport”) is equally natural and emphasizes the airport as the direct object.

What does 떠나요 mean and how is it formed?

The dictionary form is 떠나다 (to leave). To make the polite present tense you drop -다 and add -요:
떠나 + 아요 → 떠나아요 → contracted to 떠나요.
It can describe present action (“is leaving”) or a near-future event (“will leave”).

Could I say 출발해요 instead of 떠나요?

Yes. 출발하다 also means “to depart.” So 비행기가 공항을 출발해요 is perfectly correct.
Nuance: 출발하다 is a Sino-Korean verb and often used in formal announcements, whereas 떠나다 is a native-Korean verb and slightly more general.

How would I make this sentence more formal, for example in a news report?

Switch the polite -요 ending to the declarative -니다 ending:
비행기가 공항을 떠납니다.
Or use 출발하다 formally:
비행기가 공항을 출발합니다.

Can I omit 비행기가 since it’s clear from context?

Yes. Korean often drops subjects when they’re obvious. You could simply say 공항을 떠나요 if the listener already knows you’re talking about the airplane.

Why is the word order Subject-Object-Verb? Could I say 공항을 비행기가 떠나요?

Korean follows S-O-V order. Putting the verb at the end is mandatory for grammatical sentences. 공항을 비행기가 떠나요 would sound unnatural and confusing. The typical pattern is 비행기가 (S) 공항을 (O) 떠나요 (V).