The Jewish story started with Abraham. Because of his faith, Abraham's seed was promised to be numbered like counting the dust of the earth, and was forever promised the land of Israel. (Genesis 13:15-16).
This prophesy has held true as Abraham's seed (i.e. descendants)- Issac (father of the Jews) and his brother Ishmael (father of the Arabs) populate that region and still inhabit the Earth even today.
Abraham's seed was also promised a savior, and the entire Hebrew Bible comprises the Jewish history and their relation with YHWH, and the coming of this savior and his promise to restore the Earth to peace.
Jesus' birth separates these two religions.
The first Christians in history were actually Jews that 'continued' their own version of Judaism, believing that the promised savior of Abraham, Issac, Jacob, to David generation, etc., was Jesus.
Muslims on the other hand view Jesus as a prophet of YHWH, or Allah as they call God, and it ends there; whereas the traditional Jews rejected Jesus and killed him as they had a history of killing their prophets sent by YHWH (read how Isaiah was killed, for example).
600 years after Jesus' death, Muhammad was born in Mecca and started his own version of the continuation of the Abraham story above.
They saw Muhammad as God's prophet, sent by the angel Gabriel, to Muhammad to write the Koran, so that is their savior.
That's the very short version. : )