This question has bothered historians throughout the ages leading modern scholars to consider the Exodus story as nothing more than “Jewish Mythology”. They point to various archeological findings that seem to negate (in their view) the possibility of an Exodus, of the proportions described in the Torah. They postulate when the Exodus might have happened, and when they don’t find archeological or historical evidence, they categorically declare that it never happened.
I’m perfectly comfortable relying on the Torah to verify the story’s authenticity but for those more rationally minded individuals here’s a nice article that helps to clarify the record and align the biblical account of the Exodus with the picture painted by archeology: Click here to read.