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Matthew 17:3 tells us that Moses and Elijah appeared and were talking with Jesus: Just then there appeared before them Moses and Elijah, talking with Jesus. It was after Peter and James saw and heard them all talking that he suggested making the tabernacles and mentioned them by name, so it's possible that their greeting and conversation revealed the identities of Moses and Elijah. Other than that, the Bible doesn't specifically tell us how they were so easily recognized.
This isn't stated in the passage, but I've always understood that it was revealed to them by the Holy Spirit. It is possible that they were actually recognized without divine intervention. The disciples would have been raised with the history of these two prophets, and they may have been carrying some item which made them identifiable based on their roles in history (such as if Moses carried two stone tablets). However, I've typically inferred that this knowledge was simply made known to the disciples so that they could understand at least a piece of what they were witnessing.
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