Messianic Judaism is a movement of Jewish people who believe that Yeshua (Jesus' original name in Hebrew) is the Messiah of Israel and the Savior of the world. Yeshua is the most Jewish of Jews. Yeshua was a descendant of both Abraham and King David, was raised in a Jewish home and went to synagogue. He perfectly kept the entire Torah (see Galatians 4:4). He taught that He came to fulfill, not set aside, the Torah (see Matthew 5:17-19). He was a rabbi who performed unparalleled miracles, bringing great blessing to the nation of Israel.
All His early disciples also lived very Jewish lives. The Messianic movement was entirely Jewish at its inception, and continued to exist as an authentic Jewish movement for 700 years after Yeshua's death and resurrection.
Messianic Jews have not stopped being Jewish. On the contrary, we remain strongly Jewish in our identity and lifestyle!