Luke 9:30 “And behold, two men talked with Him, who were Moses and Elijah,”
The purpose of the presence of these men is not immediately clear. However, it is clear that the ultimate point of the transfiguration experience was the glory of Christ. The three disciples allowed to see it would remember it for that purpose and share it later. So, the presence of Moses and Elijah are a part of that.
How the disciples knew who these men were is also not immediately clear. Elijah might have been easily recognizable. But, it seems more likely that this knowledge came either as a result of the conversation between Jesus and the two, or (more likely) part of the miracle of the event was this prescience on the part of the observers. The response of Peter to the whole presentation would seem to support the latter conclusion as he was obviously overwhelmed by what was given to them to know and see.
One might also ask, why these two men specifically? There could be several possible answers here, though all of these could equally be the case. For one, maybe Jesus specifically chose these men because He wanted to speak to them at that particular moment. He discussed His upcoming sacrifice with them, and given the fact that they appeared in glory with Him, we have no reason to suspect they were only images of the men. It seems they two were allowed to actually be there with Him. But also, it could be that they were selected for what they represent in Scripture. Moses is the foremost in relation to the law. Elijah serves as the representative of the prophets.
And then, finally, the other reason for this selection of Moses and Elijah could be that they will be the two witnesses we are told about in the book of the Revelation. This is not certain, but has a decent support in that the two witnesses will be associated with judgments that Moses and Elijah were also associated with.
Heavenly Father, thank You for the truth of the Scriptures! You have given us so much, may we love You forever for it! You alone know the end from the beginning and have Your perfect plan in motion from day one. In the Name of the Lord Jesus Christ I pray and praise You, Amen.