How exactly do register variables work in C? When should they be used? K&R touches upon them briefly but moves right along. Is there any source in which I can read about them in depth?
No, again, in C, a variable is a named object. However, an object isn't a variable.
How do these not get lost?
Objects have scope, duration, and linkage. The guy that started NetBSD told me once upon a time to think of these three things of being the property of an object. So if the object has file duration, the life of the object will persist after the function returns.