Answer 1
It doesn't. Each religion sincerely believes that theirs is the one that is true and correct, but it is not accurate (nor fair to Christians and Jews) to say that Islam "corrects the mistakes" of these two great religions. What Islam does is offer a different set of answers to the questions people ask about the meaning of life; it also provides another path to find comfort and a closer relationship with God (Allah). Obviously, Muslims believe that Islam has the best answers, and some Muslims are undoubtedly certain that their religion is better than the others; but Jews and Christians believe their answers are equally beneficial, and they are just as proud of what their religions teach.
Answer 2
Islam certainly makes the claim that God revealed the Qur'an to Mohammed for a number of reasons including to correct the mistakes inherent in the Jewish and Christian Holy Books and correspondingly in Jewish and Christian religious practice. There are two views typically proffered for why Islam makes this claim. The first is the view from Islamic Theology and the second is the view of Skeptics and Non-Muslims.
Islamic Theology
Islam holds that God has four holy books up with Him in a pure and untouched form: the Taurat (Pentateuch or Torah), the Zabur (Psalms - although in Islam, the Zabur was revealed to David as opposed to being Divine Inspiration or the work of the Holy Spirit), the Injil (Gospel - although in Islam the Injil was a book revealed to Jesus and that he revealed as opposed to four books written by his followers), and the Qur'an. Each book was sent down to the appropriate Rasul or Messenger who gave the book to the people and revealed God's message for mankind. However, Islam asserts that Jews and Christians modified these texts over time and continue to alter them. Some say that this alteration was unintentional and is simply a result of bad copying and remembering, while others say that influential Jews and Christians purposefully altered their holy books for political reasons. As a result, God revealed the Qur'an in order to rectify these errors in the text and clarify what The Bible used to be. In this view, Mohammed is a hero for helping Jews and Christians to realize the errors in their own texts.
Skeptics and Non-Muslims
Mohammed grew up in Arabia surrounded by Jews and Christians and it is quite likely that he heard their stories and religious excerpts, especially considering that some of the lines of Qur'an are direct lifts from the Talmud. However, Mohammed got a number of details fo those stories incorrect since he was reciting what he had heard from someone else, like a game of telephone. When Jews and Christians then came to him and said that these assertions about their tradition were incorrect, Mohammed was stuck in a bind. He had claimed to have direct revelation from God and yet he had made obvious errors that God would not have made. Therefore, he revealed that the Jewish and Christian Scriptures were modified in just such a way that his revelations showed what those Scriptures ought to have been. He also had to explain why these changes had accrued and therefore also revealed that Jews and Christians had tampered with their texts. This could conceivably cover up any error that he made when discussing the Bible and so it worked as a long-term solution for Mohammed.
Islam, Judaism and Christianity.
Judaism, Christianity, and Islam have all claimed the Holy Land for themselves.
Judaism, Christianity, and Islam religions.
For Judaism and Christianity it is Jerusalem. For Islam it is Mecca.
-- Islam -- Judaism -- Christianity
Christianity and Judaism
Abraham is considered the father of Judaism, Christianity and Islam as the Old Testament's 5 books of Moses - The Pentateuch (books are: Genesis, Exodus, Leviticus, Numeri and Deuteronomy) are included in the Holy Book of Judaism that is Torah. Eventually he is considered as the father of Islam basically claims entire religions of Judaism and Christianity as "prophets" and predecessors of Islam.
The question should be " What does Christianity have to do with Jews (or Judaism)?" Judaism is like a tree where Christianity is its branch and Islam is it leaves. Christianity is based on Judaism and Islam is based on Christianity. The major difference between the three is that Judaism doesn't have jesus in it unlike Christianity and Islam.
Christianity and Islam both trace their roots to Judaism.
Islam. It claims that it is the continuEation of Judaism and CHRISTIANITY
Both Islam and Christianity originated out of Judaism.
There is only One God in Christianity, Judaism and Islam.