The Qu'uran designates Christians and Jews as "People of the Book"; that is, that Christianity and Judaism have connections to some figures that were prevalent in Islam (Abraham and Isaiah come to mind). People of the Book were given "Dhimmi" (Protected People) status, and the Qu'uran advocates for discussion and possible conversion with Christians and Jews rather than conflict due to their similar roots. This is akin to Bernard of Clairvaux in Europe strongly criticizing Christians who persecuted Jews; they all stem from the same religious roots, and the goal should be cooperation and conversion rather than bloodshed.

This is not to say that Muslims treated all non-Muslims fairly. The Qu'uran expressly forbids polytheism; pagans would not be protected by Muslims and would likely have faced greater repercussions than just paying the jizya.

tl;dr The Qu'uran (and by extension, Muslim law) designate Christians and Jews as protected people due to their common religious roots.