It's actually a very interesting story. Here goes my attempt to explain it:
If you're familiar with the Old Testament (which actually Jews, Christians, and Muslims ALL believe is Holy, but we'll get to that in a minute), you likely know who Abraham is (he's where the term "Abrahamic religion," which includes Judaism, Christianity, and Islam comes from). If you don't, he was essentially the "patriarch" of what would become Judaism. Anyway, God talks to Abraham, and Abraham listens. In Genesis (the Old Testament book), God promises Abraham that he will give him a great nation and many descendants. Abraham and his wife, Sarai/Sarah, try to have a child, but are unable to. So Sarah tells Abraham to have a child with her servant, Hagar, as they believed that this was the only way they could have a child, which they had to do to fulfill God's promise of Abraham having many descendants and a great nation. However, God later allows Abraham and Sarah to have a child together, and this child is named Isaac. As Isaac and Ishmael grow up, Sarah tells Abraham to make Ishmael leave so only her son (Isaac) will inherit the kingdom. Abraham doesn't want to do this, exile his own son, but God assures him that He will "make a nation of him also."
So what became Judaism evolved through the lineage through Abraham to his son Isaac, and Christianity also eventually developed out of this lineage. But, as God said, Ishmael would also have a nation. When Mohammad started Islam, he claimed to be a descendant of Ishmael, which means that Islam believes that they are the nation that was promised to Ishmael by the Abrahamic God. The geography of where Islam began seems consistent enough with where Ishmael traveled to. Furthermore, once Ishmael was sent away, it seems that the two sons, Isaac and Ishmael were at least not openly hostile towards each other, as they were both present at their father's funeral. Furthermore, there is actually a decent history of Muslims and Jews getting along well enough, often much better even than how the Christians treated the Jews at some points in history.
TL;DR: Abraham is the patriarch of the Abrahamic religions (Judaism, Christianity, and Islam). He has a son, Ishmael, with his servant, and later another son, Isaac, with his wife. Ishmael is sent away, but God promises to make a nation of him. Isaac goes on to continue what would become Judaism (and later also Christianity), while Muhammad, the founder of Islam, claimed to be a descendant of Ishmael, which would mean that Islam is the nation that Ishmael was promised by God.
Of course, you don't have to believe any of this, but the Old Testament is the "common thread/core/beginning" of all three religions.