It may come as a surprise to Muslims that the West originally took offense at them. Because Islam from its 7th century A.D. founding honored the Hebrew Prophets and spiritual leaders, including Jesus, Jews and Christians at first thought Islam was a schism in their own religions, rather than a new religion in its own right. Thus Jews and Christians took offense at Islam for distorting or misappropriating Jewish and Christian history and concepts.
For example, a Prophet in Islam is a leader of the temporal and religious community. In ancient Israel, however, a Prophet scandalized, rather than led the community, admonishing everyone from common folk to kings regarding their behavior. Because of this many Prophets were not revered until long after they had been killed by their contemporaries. Therefore, leaders, much less founders of religion, such as Abraham and Moses, are not called Prophets in Judaism or Christianity. In clouding this concept, Islam may devalue dissent, silencing voices that may warn it of going astray.
Muhammad, on the other hand, was a founder, a leader, and a Prophet. Naturally, he used the latter word to embrace all three roles, but this obfuscated the history of the Jews, in whose culture these roles had been distinct. Moreover, he abolished the Mitzvot, the ritual keeping of Jewish Law, and set forth a new Islamic Law which was similar in form and function, but different in content from the original. The new Law was much better suited to the culture of his people, but it is understandable that the Jews felt their history and customs had been absorbed and misrepresented.
Christians were outraged on three counts. (1) By the time of the Revelation to Muhammad, the idea of the Trinity — the Father-Son-Holy Spirit identity of the One God — had become Christian Doctrine. The Qur'an recommends against this doctrine as being too confusing, especially for newly converted polytheists, who would have misunderstood any seeming plurality of God to include some of their favorite idols. (2) The Qur'an denies the Divinity of Christ. In the Arab culture at the time, one had a son to take care of one's old age, and to inherit one's wealth. God, not having either of these problems, has no need of a son in this sense. Now, in Christian Doctrine, the idea that Jesus is God's Son means that God miraculously remained God, while simultaneously becoming born as an ordinary human baby, who grew up to live and preach among us, as one of us, with no special advantages or recognition. That is to say, contrary to all previous and subsequent ideas of God, God shed his majesty, dignity, and otherness, and went through life just like every one of us.
And, in what is a cosmic irony and a damning indictment, according to Christian tradition we responded to this by killing our God for heresy when he taught us about himself, after which God forgave us by restoring Jesus to life before taking him back to himself. This brings us to point (3). Islam teaches that the incarnation of God in Jesus is somehow beyond God's power, or beyond God's will. And further, that Jesus only appeared to die on the Cross. This is taken by Christians to be blasphemy — pretending that the suffering of the man was only simulated, that the suffering of God at our hands and on our behalf was unreal. Again, it is only natural for the Qur'an to deny Christ's suffering — the early Muslims were engaged in an armed struggle for their existence. It would have been poor judgment to teach them anything that would lead them to imagine that the mark of their Prophet's authenticity would have been for them to lose that struggle, and for him to lose his life. But in discarding the Crucifixion and the Resurrection, they also discard unique Christian insights into the human condition of Sin, Forgiveness and Redemption.
The Christians treated these doctrinal differences regarding Jesus as they treated such differences among Christians — they regarded it not as a new religion, but as a heretical form of their own religion. The Prophet was regarded as the creator of a schism among the faithful, the assumed just punishment for which was vividly illustrated in Dante's Inferno. (His 16th century depiction of the Prophet in Hell makes any offense by Salman Rushdie insignificant in comparison!)
The early Muslims, for their part, were outraged at their treatment as heretics and schismatics by the Jews and Christians. They saw themselves as following the straight path laid down before the founding of both religions, and as correcting their errors. Muslims saw the Jews and Christians as stubborn and unwilling to learn anything new or better than their old ways, because their spiritual imaginations had become captive to their Scriptures — Jews and Christians had become "People of the Book."
Now Judiasm, Christianity, and Islam are all well-established and venerable. We need to respect them as distinct but related religions, all following in the footsteps of father Abraham. They are no longer schismatic forms of one another. They are three parallel paths, ordained for three peoples.
For more details on Islam, try looking up some of these verses in the Qur'an.