Why does car become care when an e is added to the end?

This is a little like asking "Why do vehicles stop at stop signs?" There is no fundamental law of physics that causes it; for the most part, drivers simply follow the rule. It's the same here; there is no fundamental truth behind the change in pronunciation; it is simply the general rule. There are countless words in English where a vowel changes sound because of a following 'e'. Other examples similar to car: tar, mar, war, bar, far, har, par. There are deeper linguistic things going on that are beyond the intention of the question, having to do with the nearly miraculous development of the alphabet. (It seems anything but miraculous to students, I know.) We would have a very long and complicated alphabet if we insisted on a different symbol for every possible sound in a given language. We came [cam-came] up with a way to encode our spoken language with a little more efficiency and flexibility. The written word is only an approximation of the spoken word, which is itself a symbol for something else entirely.