According to the explanation of St. John of Damascus, “the Lord took Peter in order to show that His testimony truly given to him will be affirmed by the testimony of the Father and that one should believe him in His words, that the heavenly Father revealed this testimony to him (Mt. 16:17). He took James as the one who before all the Apostles would die for Christ, to drink His cup and be baptized with His baptism (Acts 12:2). Finally, He took John, as the virgin and purest organ of Theology so that he, after having beheld the eternal glory of the Son of God, has thundered these words: ‘In the beginning was the Word, and the Word was with God, and the Word was God’ (Jn. 1:1). Besides this on the Mount of Transfiguration, Peter who hadn’t yet spread the ideas about the suffering and death of his Teacher and Lord (Mt. 16:22), might mature in the truth of His glory, which forever remains inviolable despite all hostile efforts; James and John, awaiting the opening of the earthly kingdom of the Messiah and pursued the first places in this kingdom (Mk. 10:37), might behold the true majesty of Christ the Savior, surpassing every terrestrial power. The three disciples were under the law (Deut. 19:15) sufficient witnesses of the revelation of the glory of God and, according to the expression of St. Proclus, ‘in spirit personally represented all the others’.”
source: mystagogy