"Even as Islam spread beyond the borders of Arabia into the Byzantine and Sassanid Empires (Syria and Persia), non-Muslims were accorded certain rights. If they accepted the authority of the new Islamic government, then treaties were concluded and the non-Muslims paid a special tax, called a jizya."
In Arabia proper, Islam had only one choice to offer- convert or die. And indeed they annihilated the non-muslims from Arabia to the last. When they spread out of Arabia they allowed non-muslims to live under sever civil and religious repression. That included heavy tax, inequality by law, forbidding of building new churches/synagogues, and outright attacks and robbery.
"Those who believe (in the Qur'an), and those who follow the Jewish (scriptures), and the Christians and the Sabians,- any who believe in Allah (G-d) and the Last Day, and work righteousness, shall have their reward with their Lord; on them shall be no fear, nor shall they grieve." (Qur'an, 2:62)
There are many occurrences in the Quran of this kind of verse and in some of them it says something like those who accept all of the books and the prophets are good and those who only accept part of the book (meaning Jews and Christians who did not acknowledged Muhammad as a prophet) are bad...
So it is my understanding that the Quran make a claim that one can be Jew/Christian and accept muhammad's prophesies because they supposedly are a continuation of his own faith.