Tomorrow morning we're leaving Japan.

Literal

Tomorrow [possessive-の] morning, we [topic-は] Japan [object-を] depart.

発つ (たつ) means 'to depart/leave [a place]' and takes its origin with を — the same を that normally marks direct objects also marks the place you leave from (出る [room] を出る, 卒業する [school] を卒業する). Here, 'depart Japan' = 日本を発ちます. Different kanji from 立つ ('stand'), same reading.