‘Now Peter and those who were with him were heavy with sleep, but when they were fully awake (or ‘having remained awake'), they saw his glory, and the two men who stood with him.'
But meanwhile Peter and the others were heavy with sleep, as they would also be in Gethsemane (Luke 22:45-46; Mark 14:37; Mark 14:40-41), a deliberate pointer to their weakness as mere men in the face of the revelation of the divine. They were learning their insufficiency in the things of God. But at what was happening they were awakened, and once they were fully awake they saw the glory of Jesus and the two who stood with Him. Their eyes were opened to see His glory (2 Corinthians 4:4-6).
Or the verb may signify that somehow they managed to stay awake, which is the usual meaning of the verb. The idea is probably that, while they saw all, their senses were dulled. It is possibly an attempt to explain Peter's rather foolish statement.