“These are my teachings, which I told you while I was still with you ...” (Luke 24:44)

“These are my teachings, which I told you while I was still with you – that all the things written about me in the Scripture of Moses and the Prophets and the Psalms must be accomplished.” (Luke 24:44)
Jesus has appeared to his disciples after the death of his physical body. He has "risen" from that body - left the body - and now appears before their eyes in a form they could understand. Jesus is speaking of both his teachings and his example in this statement.

The Greek word, λόγος (logos) indicates Jesus is speaking of his teachings. This word means, "doctrine, teaching" according to the lexicon.

Unfortunately, Luke doesn't reiterate those teachings in this verse as Jesus said this. These were recorded throughout the Gospel of Luke. Instead, we find the verse following:
Then he opened their minds to understand the Scriptures. (Luke 24:45)
Thus we find that Jesus allowed them to understand the core teachings of the Scriptures. Whether or not he reiterated those teachings at the time, or whether he simply made them aware of what he had taught them before doesn't matter at this point. What matters is that Jesus is referring to his teachings and the example of his life with reference to the teachings of the Prophets and the Scriptures at that time - which were the teachings of the Prophets.

Many interpret this to simply mean the Prophets predicted Jesus' appearance and persecution.

But this is a severely limited view of Jesus' life and teachings. This sectarian version of the meaning of Jesus' life completely ignores his very teachings.

Such a view - that Jesus' life was only about his persecution - ignores Jesus' entire mission.

It ignores the fact that Jesus traveled around the hot and dry region of Judea for years, teaching thousands of people about God. His primary teaching, in fact, was based upon Moses' teaching:
“ ‘Love the LORD your God with all your heart and with all your soul and with all your mind.’ This is the first and greatest commandment." (Matt. 22:37-38)
This statement is a direct quote from Moses' teachings - Deuteronomy 6:5. Why is this important? Because this is what Jesus considered to be the most important instruction - to love God with all our heart.

Was this a passing statement by Moses? Did Moses not take this instruction seriously? Absolutely not. Moses repeated this instruction over and over to his students:
Love the LORD your God with all your heart and with all your soul and with all your strength. (Deut. 6:5)
Love the LORD your God and keep His requirements, His decrees, His laws and His commands always. (Deut. 11:1)

So if you faithfully obey the commands I am giving you today—to love the LORD your God and to serve Him with all your heart and with all your soul— (Deut. 11:13)

If you carefully observe all these commands I am giving you to follow—to love the LORD your God, to walk in obedience to Him and to hold fast to Him— (Deut. 11:22)

"... because you carefully follow all these laws I command you today—to love the LORD your God and to walk always in obedience to Him—" (Deut. 19:9)
"For I command you today to love the LORD your God, to walk in obedience to Him, and to keep His commands, decrees and laws; then you will live and increase, and the LORD your God will bless you in the land you are entering to possess." (Deut. 30:16)
"... and that you may love the LORD your God, listen to His voice, and hold fast to Him. For the LORD is your life, and he will give you many years in the land he swore to give to your fathers, Abraham, Isaac and Jacob." (Deut. 30:20)
Then we discover that Moses' student, Joshua, also passed on Moses' teaching:
But be very careful to keep the commandment and the law that Moses the servant of the LORD gave you: to love the LORD your God, to walk in obedience to Him, to keep His commands, to hold fast to Him and to serve Him with all your heart and with all your soul.” (Joshua 22:5)
So be very careful to love the LORD your God. (Joshua 23:11)
Then we find that David also passed on this clear instruction:
"Love the LORD, all His faithful people!" (Psalms 31:23)
Over and over we find similar instructions from other Prophets - some mistranslated to "fear" from the Hebrew word יָרֵא (yare') - which means, according to the lexicon, to "revere" or "to stand in awe of, be awed" and "to cause astonishment and awe, be held in awe." 

Someone who loves God will also revere Him and hold Him in awe.

In fact, loving the Supreme Being is the common teaching among all of the Prophets. While they might not all quote Moses' instruction per se, it is obvious by their lives and examples that they loved the Supreme Being, and taught this to others.

The quotes above - along with the activities of the Prophets - indicate that love of God was not exclusively Jesus' teaching. Love for God is the basis - the primary fundamental of the teachings of all of God's representatives.

And this is the primary reason why Jesus is referring to "all the things written about me in the Scripture of Moses and the Prophets."

Jesus is the embodiment of the teachings of the Prophets. How so?

First, because Jesus' mission was focused on teaching his students to love and serve the Supreme Being.

Second, because he was personally dedicated to the Supreme Being. He loved and served God with his very life.

Just think for a second about Jesus' life. He didn't have a family or career. He didn't have a house. He didn't save any money. He lived only for God. He walked around the countryside barefoot, teaching people about love for God. He completely dedicated his life to the Supreme Being.

Then, of course, he was persecuted for his teachings and dedication to God. His mission threatened the authority of the Jewish high priests and the institution itself. So they murdered his physical body.

So not only did Jesus live for the Supreme Being, but he accepted the murder of his physical body as part of his service to the Supreme Being. This is what many consider the greatest sacrifice. To give the life of one's physical body for the sake of someone else. In this case, Jesus gave the life of his physical body for the Supreme Being. And for all of those who come to understand this.

These activities - Jesus' teachings and his life and persecution - embodied the teachings and example of the Prophets. Whether or not Jesus' persecution was specifically predicted by a Prophet doesn't really matter. What matters is that Jesus' teachings and example embodied their teachings.

One could argue, as many have, that the prophesies of Isaiah, Micah and/or Zechariah were not predictions of Jesus per se. That they were predictions of a specific king that was to come, or a prediction of another teacher who would be persecuted (as the Jews persecuted a number of Prophets and teachers over the centuries). Or in some cases, that Jesus could have orchestrated a particular activity because it reflected something one of the Prophets said.

Again, it doesn't matter, because Jesus' life and teachings overshadow these minute details. After all, what does it matter to us whether Jesus' arrival was predicted by the Prophets? It doesn't prove anything one way or another. Many simply want to prove their sect is superior, or that somehow Jesus is more superior than previously thought. Some of these are the same people that focus upon Jesus' miracles - as though they need Jesus' miracles to prove to others that their own teachings about Jesus are justified.

Jesus himself criticized this type of sectarian puffing created by some who claim to follow him:
"Not everyone who says to me, ‘lord, lord,’ shall enter the sanctuary of the spiritual realm – only one who does what pleases my LORD in the spiritual realm. Many will say to me at that time, ‘Master, did we not prophesy in your name and in your name cast out demons, and in your name perform many miracles? And I will say to them, ‘I never knew you: Get away from me, you who practice wickedness.’ Thus, anyone who hears these teachings of mine and acts on them is like a wise man who built his house on a rock. Then the rain fell, and the floods came, and the winds blew and beat against that house – and yet it didn’t fall because it was founded upon the rock. And anyone who hears these teachings of mine and does not act on them is like a foolish man who built his house on the sand: The rain fell and the floods came and the winds blew and beat against that house, and it fell – it fell with a great crash.” (Matt. 7:21-27)
So we find with this clear statement of Jesus that pleasing God - doing God's will - is what Jesus wants from his followers. He doesn't want their grandstanding about miracles and casting out demons. He doesn't want their praises and glorifications in order to prove their religiousness. Jesus wants his followers to do what he taught us: Come to love and serve the Supreme Being with all our heart and soul.