Jesus fulfilled six of the seven festivals in a jubilee year. And, the seventh festival (the Festival of Tabernacles) points to the second coming of Jesus.
He didn't; the claim is spurious. However, if we look at the Christian claim he may have fulfilled some of the festivals, we then see the fulfillment citations are tenuous at best.
The first four of the seven feasts occur during the springtime (Passover, Unleavened Bread, First Fruits, and Weeks), and they all have already been fulfilled by Christ in the New Testament. The final three holidays (Trumpets, the Day of Atonement, and Tabernacles) occur during the fall, all within a short fifteen-day period.
Many Bible scholars and commentators believe that these fall feasts have not yet been fulfilled by Jesus. However, the “blessed hope” (Titus 2:13) for all believers in Jesus Christ is that they most assuredly will be fulfilled. As the four spring feasts were fulfilled literally and right on the actual feast day in connection with Christ's first coming, these three fall feasts, it is believed by many, will likewise be fulfilled literally in connection to the Lord's second coming.https://www.gotquestions.org/Jewish-feasts.html
1) Passover (Leviticus 23:5) – Pointed to the Messiah as our Passover lamb (1 Corinthians 5:7) whose blood would be shed for our sins. Jesus was crucified on the day of preparation for the Passover at the same hour that the lambs were being slaughtered for the Passover meal that evening (John 19:14).
2) Unleavened Bread (Leviticus 23:6) – Pointed to the Messiah's sinless life (as leaven is a picture of sin in the Bible), making Him the perfect sacrifice for our sins. Jesus' body was in the grave during the first days of this feast, like a kernel of wheat planted and waiting to burst forth as the bread of life.
3) First Fruits (Leviticus 23:10) – Pointed to the Messiah's resurrection as the first fruits of the righteous. Jesus was resurrected on this very day, which is one of the reasons that Paul refers to him in 1 Corinthians 15:20 as the “first fruits from the dead.”
4) Weeks or Pentecost (Leviticus 23:16) – Occurred fifty days after the beginning of the Feast of Unleavened Bread and pointed to the great harvest of souls and the gift of the Holy Spirit for both Jew and Gentile, who would be brought into the kingdom of God during the Church Age (see Acts 2). The Church was actually established on this day when God poured out His Holy Spirit and 3,000 Jews responded to Peter's great sermon and his first proclamation of the gospel.
5) Trumpets (Leviticus 23:24) – The first of the fall feasts. Many believe this day points to the Rapture of the Church when the Messiah Jesus will appear in the heavens as He comes for His bride, the Church. The Rapture is always associated in Scripture with the blowing of a loud trumpet (1 Thessalonians 4:13-18 and 1 Corinthians 15:52).
6) Day of Atonement (Leviticus 23:27) – Many believe this prophetically points to the day of the Second Coming of Jesus when He will return to earth. That will be the Day of Atonement for the Jewish remnant when they “look upon Him whom they have pierced,” repent of their sins, and receive Him as their Messiah (Zechariah 12:10 and Romans 11:1-6, 25-36).
7) Tabernacles or Booths (Leviticus 23:34) – Many scholars believe that this feast day points to the Lord's promise that He will once again “tabernacle” with His people when He returns to reign over all the world (Micah 4:1-7).https://www.gotquestions.org/Jewish-feasts.html
This case does not seem to be built on very strong evidence. It looks like speculation; there is no support for the conclusions drawn. First, at best you can say 4 out of 7 were 'fulfilled', but if you look closely at 4 this is not when Jesus said he was establishing his church; it is just an event that happened on that day and which was written in the bible. Furthermore Jesus wasn't present to 'fulfill' the feast (whatever that means; attending it, I suppose?) so we're down to 3 out of 7. Two is a stretch, any righteous person even Mary could have been said to 'fulfill' such a feast, but what this really points to is the strange conclusion that a feast could be 'fulfilled' in the first place.
Also known as Sukkhot, or Booths.
And the feast of harvest, the firstfruits of thy labours, which thou hast sown in the field: and the feast of ingathering, which is in the end of the year, when thou hast gathered in thy labours out of the field.Exodus 23:16 (KJV)
22 “You must observe the Feast of Weeks—the firstfruits of the harvest of wheat—and the Feast of Ingathering at the end of the year.Exodus 34:22 (KJV)
24 “Tell the Israelites, ‘In the seventh month, on the first day of the month, you must have a complete rest, a memorial announced by loud horn blasts, a holy assembly. Leviticus 23:24 (KJV)
The Feast of Tabernacles is described in the Old Testament. This particular feast was celebrated for seven days (Lev. 23:40). Furthermore, the first day of the festival and the day after the feast (i.e. the eighth day) were special Sabbaths (Lev. 23:39), “marking not only the climax of the religious year but symbolizing the rest of the believer in his God (Lev. 23:39).” The Feast of Tabernacles was to be a joyous celebration (Lev. 23:40) occurring immediately after the olive and grape harvests in the land of Israel. After the Israelites had gathered in the fruit of these harvests, they would construct huts from “the fruit of splendid trees, branches of palm trees and boughs of leafy trees and willows of the brook” (Lev. 23:39-40) to inhabit for the duration of the feast (Lev. 23:42). In addition to the construction of ‘foliage’ huts, Israel was commanded to offer extensive sacrifices throughout the feast:
The feast was one of the most expensive and lavish of the year, celebrated in gratitude for the people’s freedom from slavery. In the temple the offering for the first day was thirteen bulls, two rams and fourteen sheep. Each day thereafter the number of bulls was reduced by one. The total offering was 71 bulls, 15 rams, 105 lambs and 8 goats. By eating God’s food in a state of purity the people enjoyed real fellowship with their God.McFall, “Sacred Meals,” 753.
Leviticus 23:42-43 describes the purpose of this festival: “You shall dwell in booths for seven days. All native Israelites shall dwell in booths, 43that your generations may know that I made the people of Israel dwell in booths when I brought them out of the land of Egypt: I am the LORD your God.” The festival commemorated Yahweh’s gracious provision not only during the wilderness wanderings but also the recent harvests.
The Christians also claim that Jesus is the Tabernacle.
7 And she brought forth her firstborn son, and wrapped him in swaddling clothes, and laid him in a manger; because there was no room for them in the inn.
8 And there were in the same country shepherds abiding in the field, keeping watch over their flock by night.
9 And, lo, the angel of the Lord came upon them, and the glory of the Lord shone round about them: and they were sore afraid.
10 And the angel said unto them, Fear not: for, behold, I bring you good tidings of great joy, which shall be to all people.
11 For unto you is born this day in the city of David a Saviour, which is Christ the Lord.Luke 2:7-11 (KJV)
Why was there no room at the inn? Bethlehem is only about 5 miles from Jerusalem, and all the men of Israel had come to the Temple in Jerusalem to attend the festival of Booths/Ingathering/Tabernacles as required by the law of Moses (Exo. 23:14-17, 34:22-23, Deut 16:16). Every room for miles around Jerusalem would have been already taken by pilgrims, so all that Mary and Joseph could find for shelter was a stable. During Tabernacles, everyone was to live in temporary booths (Sukkot), as a memorial to Israel's pilgrimage out of Egypt - Lev. 23:42-43. The birth of the Savior, in what amounted to a temporary dwelling rather than a house, signaled the coming deliverance of God's people from slavery to sin, and their departing for the promised land, which is symbolized by Tabernacles.
Also of note is the fact that the Feast of Tabernacles is an eight day feast (Lev 23:36, 39). Why eight days? It may be because an infant was dedicated to God by performing circumcision on the eighth day after birth:
Luke 2:21 And when eight days were accomplished for the circumcising of the child, his name was called JESUS, which was so named of the angel before he was conceived in the womb.
So the infant Jesus would have been circumcised on the eighth and last day of the Feast of Tabernacles, a Sabbath day. The Jews today consider this a separate festival from Tabernacles, and they call it Shemini Atzeret.