Good question, here are some basic reason along with witnessing tools:

The Tanach [(composed of the Torah, the Nevi'im, and the Ketuvim) Major and Minor Prophets and Wisdom books] is transparent on the subject of the role of the messiah to them. It should be noted that, although there are many sections throughout the Tanach that vividly describe how the world will be forever transformed with the arrival of the messiah, very few are about the messiah personally. The vast quantity of messianic scripture in the Tanach (Torah, Nevi'im, and Ketuvim) depicts the state of perfection that the world will achieve at the end of days. It is quite clear from the vantage point of the Tanach that the significance of the messiah himself pales in comparison to the utopian age that his arrival will usher in. In Jeremiah, chapter 33, verse 17, G-d says that the royal House of David will never lack a man to sit on the throne of Israel.

The reason Jews and Noahides (Righteous Gentiles) don't accept the messiah of Christendom is because Yeshua (Jesus) did not fulfill any of the clear messianic prophecies foretold in the Tanach. In addition, the Tanach never tells of believing in the messiah because either the events leading to his advent will be so undeniable, or his reign will be a historically verifiable reality and they believe self-evident to anyone. Because no person has ever fulfilled the prophecies in the Tanach given of this future King, the Jewish people still await the coming of the messiah. All past Messianic claimants, of which there has been many since the time of Yeshua and including Yeshua (Jesus), have ultimately been rejected by the Jewish people simply because they did not measure up to the prophecies. And, as we know fullfilling for prophecies by Saul (Paul) in [Romans 9-12].

Realizing that Yeshua (Jesus) did not fulfill the requirements of the messiah as outlined in the Tanach, after the resurrection the apostles quickly developed the scriptural support known as the "second coming teachings." That teach that Yeshua (Jesus) will return to the earth to "judge mankind, and quicken the dead." However, the Jews believe that there is no clear mention of a second coming in the Torah and the Tanach of this happening; blinded for our sakes. In fact, there is definitive evidence in the Christian Gospels to the contrary. And as many Jews have studied the Christian Bible, they conclude that Yeshua (Jesus) himself knew of no such returning, as in Mark 9:1, Yeshua tells his disciples "Verily I say to you that there be some of them who stand here, which shall not taste of death until they have seen the kingdom of G-d come with power." So they believe that Yeshua actual fulfilled the verses of the "false messiah" (described in Deuteronomy 13:2-6). They see this as an accurate description of Yeshua, and no second coming was ever planned by him. Remember, that there were two clear forms of Judaism in the days of Yeshua: Sadducess and Pharisees and they were divided on whether there was the resurrection or not as mentioned in the Tanach. The former seem to have been more ready to believe in Yeshua as we see in Niocodemus coming to Yeshua by night in [John].

The following is an overview of the most distinct messianic prophecies as outlined in the Jewish scriptures that will help you when witnessing -- which both Christians and Jews agree are messianic:

"See, a time is coming -- declares the Lord -- when I will raise up a true branch of David's line. He shall reign as king and shall prosper, and he shall do what is just and right in the land." -- Jeremiah 23:5 (See also Ezekiel 34:23-24, 37:21-28; Isaiah 11:1-9; Jeremiah 30:7-10; Jeremiah 33:14-16; Hosea 3:4-5)

"Lo, I will send the prophet Elijah to you before the coming of the awesome [the messiah], fearful day of the Lord. He shall reconcile parents with children and children with their parents, so that, when I come, I do not strike the whole land with utter destruction." -- Malachi 3:23-24

"And he [the messiah] shall judge among the nations, and shall rebuke many people; and they shall beat their swords into plowshares, and their spears into pruning hooks. Nation shall not lift up sword against nation, neither shall they learn war any more." -- Isaiah 2:4

"He [the messiah] will destroy death forever." -- Isaiah 25:8

"Then the inhabitants of the cities of Israel will go out and make fire and feed them with the weapons -- shields and bucklers, bows and arrows, clubs and spears; they shall use them as fuel for seven years." -- Ezekiel 39:9

"For the earth shall be full of the knowledge of the Lord, as the waters cover the sea." -- Isaiah 11:9

"And no longer shall one teach his neighbor or shall one teach his brother, saying, 'Know the Lord, for they shall all know Me, from their smallest to their greatest' says the Lord." -- Jeremiah 31:33

"All who survive of all those nations that came up against Jerusalem shall make a pilgrimage year by year to bow low to the King Lord of Hosts and to observe the feasts." -- Zechariah 14:16

"And the Lord shall be king over all the earth: in that day shall there be one Lord, and His name one." -- Zechariah 14:9

"Thus said the Lord of Hosts: 'In those days, ten men from nations of every tongue will take hold -- they will take hold of every Jew by a corner of his cloak and say, 'Let us go with you, for we have heard that G-d is with you'." -- Zechariah 8.23

Now this is the critical part of the answer to your question. They have no other reason but this verse that follows to believe that this man of perdition would be the Messiah; "And I will set My sanctuary in the midst of them for evermore. My temple also shall be with them. Yes, I will be their G-d and they shall be My people. And the heathen shall know that I the Lord do sanctify Israel, when My sanctuary shall be in the midst of them for evermore." -- Ezekiel 37:26-28 (See also Ezekiel 40-48; Isaiah 33:20). But, without their blindness to this point in History this Chrisitianity would only be another Jewish sect and the world of the Gentiles would have remained all pagen and you would not have Yeshua as your Messiah. So to the Jews first [Romans 9:32,33] Wherefore? Because they sought it not by faith, but as it were by the works of the law. For they stumbled at that stumblingstone; As it is written, Behold, I lay in Zion a stumblingstone and rock of offense: and whosoever believeth on him shall not be ashamed. So to fulfill prophecy they were blinded. And to not give up your witness to them in another place it is prophesied that G-d shall save a remnant of His people, so as a nation they will all see salvation in one day. Again (Romans 10,11). As there blindness will be lifted. They will also fulfill the 144,000 in Revelation for the remnant.