Luke 24:27 Beginning with Moses and all the Prophets, he explained to them what was said in all the Scriptures concerning himself.
Other New Testament Scripture contains the answer
To see what Jesus probably mentioned concerning himself, we should look at what his disciples mentioned later concerning the same thing.
Remember, chapter and verse did not appear until about 1,000 years later. So, at that time, a "reference" would have been made via quote or paraphrase. Also, the passage doesn't clearly list all disciples present, but since they were all in contact, whatever Jesus taught on the Road to Emmaus was likely shared with all other disciples. Thus, we can look at all disciples to get an answer.
1. Start with the passage, including the statement just before:
Luke 24:25-27 (NASB)
25 And He said to them, “O foolish men and slow of heart to believe in all that the prophets have spoken! 26 Was it not necessary for the Christ to suffer these things and to enter into His glory?” 27 Then beginning with Moses and with all the prophets, He explained to them the things concerning Himself in all the Scriptures.
This seems like quite a long conversation. And, it happened on a road traveling from one city to another. It wasn't a five minute chat. So, it could have included quite a lot of Scripture. In that, we know surely he included both Moses and the prophets.
Conclusion 1: Any Christological reference to Christ in the Prophets would be a included candidate, especially obvious references, but not only.
...That leads to Isaiah, a more obvious reference...
2. Consider what one of his disciples wrote quite plainly, most directly relating to the statement in v26 about the suffering Messiah.
1 Peter 2:24 (NASB)
24 and He Himself bore our sins in His body on the cross, so that we might die to sin and live to righteousness; for by His wounds you were healed.
This is a strong reference to...
Isaiah 53:5 (NASB)
But He was pierced through for our transgressions,
He was crushed for our iniquities;
The chastening for our well-being fell upon Him,
And by His scourging we are healed.
Conclusion 2: Since Peter describes nearly the same concept, it is likely that Jesus had elaborated on his statement in Luke 24:26 with Isaiah 53:5.
...Since Peter, in his letter, gives a near quote of such an obvious Old Testament reference, let's look at the first time he went back to the Old Testament to talk about Christ...
3. Consider Peter's speech at Pentecost
In Acts 2:14-40, Peter gives a sermon that includes many references, let's just look at one reference to Joel...
Acts 2:17 (NASB)
‘And it shall be in the last days,’ God says,
‘That I will pour forth of My Spirit on all mankind;
And your sons and your daughters shall prophesy,
And your young men shall see visions,
And your old men shall dream dreams;
Joel 2:28 (NASB)
“It will come about after this
That I will pour out My Spirit on all mankind;
And your sons and daughters will prophesy,
Your old men will dream dreams,
Your young men will see visions.
The Holy Spirit promise was also recorded by Luke (Luke 3:16; 11:13, Acts 1:5, 8), so Joel 2 is not beyond the scope of what Jesus likely mentioned on the Road to Emmaus.
Conclusion 3: Surely he at least mentioned Joel 2.
4. Consider Stephen's speech in Acts
Stephen's speech is rather lengthy and gives time to Moses, which was mentioned in Luke 24:27. The mention of "Moses" in Luke 24:27 is not a reference only to Moses as a character in Exodus, but all the books Moses is considered to have written, including Genesis.
Conclusion 4: Luke 24:27 "beginning with Moses" agrees with Stephen's reference in Acts 7:2 to [then] Abram's call, starting as far back as Genesis 12:1.
5. The entire Book of Hebrews is telling
- It is a cross-reference candidate
- It is a likely accumulated trend
Hebrews has the most elaborate, concise, and pure expansion on Old Testament prophecy and history concerning Soteriology and Christology (the work and person of Christ). Hebrews is that and that is Hebrews.
The only other pure-only conversation topic mentioned in the New Testament, where a situation was only-all about the Old Testament background pointing to Jesus of Nazareth as the Messiah, and that the Messiah must suffer, is your question: Luke 24:25-27, whatever it was Jesus explained on the Road to Emmaus.
But, Peter had also given a kind of "Old Testament survey" sermon, as had Stephen. Theologically, Paul walks through history, beginning with reference to creation in Romans 1:20 (see NASB) "For since creation..."
Throughout the New Testament after the Gospels, we see a trend of historical recaps. The first of those was Jesus on the Road to Emmaus.
Jesus starts by saying that the disciples couldn't figure it out on their own—what the prophets had already spoken (Luke 24:25). Therefore, all later New Testament references to Old Testament prophecy pointing to Christ would have started on the Road to Emmaus.
It is also worth mention that Luke wrote both the Road to Emmaus account and the sermons in Acts 2 and 7. So, in a sense, Acts 2 and 7 are the heavier, first, and most direct answers, or at least, expansions on this great question arising from what Luke wrote about in Luke 24. Still, it sets a New Testament trend that doesn't stop with Luke's reporting, but only continues in greater and greater detail and elaboration.
Conclusion 5: All New Testament reference to Old Testament prophecy about Christ followed after (and was therefore likely to have been inspired by passages Jesus mentioned on) the Road to Emmaus.
In good Bible hermeneutics, we look at "first mention". The Road to Emmaus is the first mention of Old Testament Christological prophecy in the rest of the New Testament. So, your most basic answer is: the New Testament Epistles.
We do not know "specifically" what particular verse or verses He explained to them. The text says, "And beginning with Moses and with all the prophets, He explained to them the "things" concerning Himself in all the Scriptures."
What things? I believe His role as the Christ He explained those texts which would shed light upon His coming mission. Notice at verse 32 that their hearts were burning while He was explaining the Scriptures.
Notice from verse 25 Jesus calls them foolish and slow of heart but after they recognized Him (vs31) I believe the hardness of their hearts melted. I also believe they had a doubting Thomas moment. Verse 34, "saying, The Lord has REALLY risen, and has appeared to Simon/Peter."
In short, Jesus reaffirmed who He was in the Old Testament and in the New Testament the disciples could say like Thomas, "The Lord of me and the God of me." (John 20:28). It should also be noted that at other places in the New Testament others have explained the Scriptures. Acts 17:2,11, Acts 18:24-28.