Muslims consider Jews and Christians to be “People of the Book” because they believe that the Jewish and Christian scriptures were inspired by God in much the same way that the Koran was. They honor the Jewish and Christian prophets and consider them to be important prophets in Islam as well. For this reason, they believe that the Jews and the Christians are part of the same religious tradition that Muslims belong to.
Muslims believe that the Jewish prophets and Jesus were all divinely inspired. They accept that Jews and Christians are monotheistic and that the God that all three religions worship is the same God. They believe that God first revealed himself to Abraham. This made Abraham the first Muslim. While Muslims do respect Jews and Christians as People of the Book, they also believe that these faiths have drifted away from God’s true word. They believe that the older religions became imperfect and that God revealed himself (through his angel) to Muhammad to get people back on the right track. They believe that Muhammad was God’s final prophet and that the revelations given to him are the ultimate indicators of what God wants.
Muslims, then, believe that Jews and Christians are “People of the Book” because they are all part of the same religious tradition with and they all revere the same God. The scriptures that the Jews and Christians believe in are also holy to Muslims, but Muslims believe that the revelations given to Muhammad and written down in the Koran supersede and perfect the Jewish and Christian scriptures.