Learn Hardcore Turkish: Paket eksik geldiğinden şubeye başvurdum. - I went to the branch because the package arrived with missing items.

Eksik means “incomplete” or “short.” Here it’s used predicatively with the intransitive verb gelmek to describe the manner of arrival. So paket eksik geldi literally means “the package arrived incomplete.”

What exactly is geldiğinden and how is the suffix -ğinden formed?

Geldiğinden is a nominalized verb form plus the ablative case, used to express “because.” Formation steps:
1) gel- (come) + -di (past tense) → geldi
2) geldi + nominalizer -ği → geldiği (“that it came”)
3) geldiği + ablative -nden → geldiğinden (“because it came”)

Why isn’t it eksik paket geldiğinden?

Attributive eksik paket would mean “a missing package” (i.e. the whole package is missing). To say “it arrived incomplete,” you need the verb phrase paket eksik geldi. Adjective placement changes meaning.

What case is şubeye, and why is this case used?

Şubeye is şube (branch) with the dative suffix -ye, meaning “to the branch.” Verbs like (to apply/turn to) take the dative to indicate the entity you address or apply to.