Why is this expression (A and ¬A |= C) entailed?
The statement $X\vDash Y$ means "every assignment to the variables that makes $X$ true also makes $Y$ true." Or to put it another way, "There is no assignment of variables that makes $X$ true but fails to make $Y$ true." Well, there's no assignment of variables that makes $A\land\neg A$ true, so there's certianly no assignment that makes $A\land\neg A$ true and also makes $C$ false.  So $A\land\neg A\vDash C$ is a true statement.

False entails anything is a rule, analogous to false implies anything.