The letters $A,B$, and $C$ stand for propositions, each of which may be either true of false. The point of the quoted statement is that it doesn’t matter exactly what propositions they are or which of them (if any) are true: the displayed statement is true no matter what.
You can prove it with a truth table. However, you don’t actually need one. $$(A\iff B)\lor(A\iff C)\lor(B\iff C)$$ is true if and only if at least one of the disjuncts $A\iff B$, $A \iff C$, and $B\iff C$ is true. $A\iff B$ is true exactly when $A$ and $B$ have the same truth value: both are TRUE, or both are FALSE. Similarly, $A\iff C$ is true exactly when $A$ and $C$ have the same truth value, and $B\iff C$ is true exactly when $B$ and $C$ have the same truth value.
Since there are only two truth values in classical propositional logic, no matter how you assign truth values to $A,B$, and $C$, two of them have to get the same truth value. This means that under any assignment of truth values to $A,B$, and $C$, at least one of the three disjuncts has to be true, and hence the whole proposition must be true as well.