The movement isn't new, but the resurgence of something very old. It was present in the land of Israel during the time of our Messiah Yeshua/Jesus Christ, and for a few hundred years after His death and resurrection. Messianic Jews were eventually assimilated into the organized Church and lost their Jewishness. As the Gentiles grew in number, and became the majority, they brought into this Jewish faith pagan practices.
The modern roots of the Messianic Movement began in East Europe in the 1800's and into the twentieth century. This movement was almost eliminated because of the Holocaust during World War II. In the 1970's this movement was revived, particularly in North America.
Messianic Jews today identify themselves as Jews and maintain a Jewish life-style while believing in Messiah Yeshua as the first believers in Yeshua had done. Non-Jewish believers also recognized a Jewish life style for them as well, not to claim to be Jewish themselves, but for a single new humanity to make peace and recognize that G-d has a biblical lifestyle for His redeemed people, Jew and Gentile who share in the same rich root of the olive tree. There is a realization that G-d does not have a double standard for his redeemed community. Their is a realization that the Bible contains G-d's Instructions on how Jew and Non-Jew should live out their daily lives as well as Instruction how we are to find G-d's Salvation to bring us to a Holy G-d. This is a movement of the Holy Spirit among the Jewish people as well as the non-Jewish people today as G-d brings all things to its fullness in these latter days. This is what the Ruach HaKodesh/Holy Spirit is revealing to people, this is the Messianic Movement.