Athanasius's sentence, written in the fourth century while he was defending the full divinity of Christ against Arianism, is the compressed form of the whole doctrine, and the argument runs backwards from salvation to christology. If what happens to human beings in salvation is that they are drawn into the life of God, then the one who effects it must himself be God; a creature, however exalted, could not confer what it does not possess. Theosis is thus the reason Eastern theology insisted on Nicaea, and not a later mystical addendum to it.
The scriptural anchor is the Second Letter of Peter, which speaks of becoming partakers of the divine nature, together with the line from the eighty-second Psalm that John's gospel puts on Jesus's lips, I said, you are gods. Irenaeus, Athanasius, Gregory of Nyssa and Maximus the Confessor all state the theme, and the last works out its christological logic in detail: because in Christ divine and human wills coexist without competition, human nature is not diminished by union with God but perfected.
The obvious danger is pantheism, and the Eastern answer was given definitively by Gregory Palamas in the hesychast controversies of the fourteenth century. Palamas distinguished God's essence, which is utterly inaccessible and in which no creature ever shares, from God's energies, his uncreated activity and self-communication, which are truly God and in which the creature genuinely participates. The deified person therefore becomes by grace what God is by nature, and the distinction between creator and creature is never crossed. Palamas was defending the hesychast monks who claimed to see the uncreated light of Tabor in prayer; his opponent Barlaam held that any such vision would have to be created, and the councils of the 1340s decided for Palamas.