This is a metaphor revealing the mystery of the divine activity: Father cannot purge anything unless this is on the Vine - Christ: if a branch falls from the Vine/Christ, Father is totally impotent to vivify this branch by Himself alone, outside the Vine, but can nourish the branch only through the Vine/Christ. This means, that the salvific divine activity of the Father and the Son (Jesus Christ) is and can be conducted only together and jointly, which is a clearest and irrefutable proof of divinity of the Son/Logos, called since Incarnation also Jesus Christ.
Now, the question "how the Father purges branches", is the same as to keep in mind that a branch is, although essentially also Vine (in the sense that the Vine/Christ has fulness of the human essence after the Incarnation), still branches denote and stand for created personalities, whereas the Person of the Vine/Christ is uncreated, born from the Father in Eternity, possessing the same alwaysness as possessed by the Father, being the co-God with the Latter (John 1:1), thus even if the branches can fall, the Vine always remains unfallen and faithful, possessing the same divine immunity from fall as possessed by the Father, for God does not and ontologically cannot deny His God-ness (cf. 2 Tim. 2:13). Thus, the Vine/Christ is Perfect as the Father is Perfect, while we, branches are not perfect, but in a position of a constant growth in perfection, as commanded by Christ (Matthew 5:48). If the Lord finds faults, that is to say, imperfections even in His holy bodiless angels (Job 15:15), what can be said of us who have so many imperfections, "as many, exactly, as are the waves in the ocean", as St John Chyrsostom puts it.
Vine is a symbol of the Incarnate God, - God who adopted human nature. Could God save humanity without His being incarnated or inhumanated? No way! Because salvation means ultimately that human being enters divine life, life of God, and unless God provides humans this level of intimacy by Himself becoming human (cf “vine”) no such access to divine life would be possible to humans, not even to the greatest prophets. Thus Father necessitates His Son’s Incarnation for fulfilling His love towards humans (cf. John 3:16).
Exactly this is when the branches are purged: they abandon their imperfection and embrace greater and greater degree of perfection through a joint graceful action of the Father and the Son (and Holy Ghost for that matter, but for parsimony of this discourse I do not talk more about the Holy Ghost) and the co-action or synergy of the branch, and this process has no end, for we never end growing in perfection, always stretching out to the yet unknown degrees of glory (Philippians 3:13-14); and thus branch differ from branch with regard of the degree of the glory achieved (cf. 1 Cor. 15:41).