Intuition: Why is the biconditional true if both statements are false? I already know that a false statement implies anything. Because I ask only for intuition, please do NOT prove this or use truth tables (which I already understand).
Source: p 333, A Concise Introduction to Logic (12 Ed, 2014), by Patrick J. Hurley

The truth table shows that the biconditional is true when its two components have the
  same truth value and that otherwise it is false. These results are required by the fact
  that $P ≡ Q$ is simply a shorter way of writing $(P ⊃ Q) \wedge (Q ⊃ P)$. If P and Q are either both
  true or both false, then $P ⊃ Q$ and $Q ⊃ P$ are both true, making their conjunction true. ...

I already understand the above, but am seeking an even more intuitive explanation.
 A: Consider the sentence:

I will jump off the cliff if and only if you do it as well

You did not jump off the cliff - so why should I?
The moral of the story is that a biconditional statements only states that $\alpha$ holds whenever $\beta$ is the case, they are, say, 'logically attached'. In the cliff analogy, they either jump 'together' or not.
A: In law, it is sometimes the case that a contract is binding if, and only if, both parties signed it. So, if $P$ stands for "the contract is binding" and $Q$ stands for "Both parties signed the contract", then $P\iff Q$ stands for "the contract is binding if, and only if, both parties signed it". Now, when would the claim $P\iff Q$ be true in a particular situation? If both parties signed the contract and the contract is binding, then $P\iff Q$ is true. If not both parties signed it, nor is the contract binding, then $P\iff Q$ is still true. In any other case it is false. 
A: To adapt the example given in the link, consider "I will give you $1000000 if and only if pigs fly".
It means that I am obligated to give you \$1000000 if pigs fly, and that pigs must exist if I am to give you \$1000000. Just actively see it (at least at first) as two conditionals; work out a concrete causal picture and then find a way to invert the direction so that it makes sense.
A: "$\Leftrightarrow$" and "$\leftrightarrow$" have completely different meanings in mathematics. The first one means what is on the left is equivalent to what is on the right. However, the second one is what you are looking for as bidirectional and says nothing about the equivalency.
So if I say that "pigs can fly if and only if the moon is made up of cheese", what is your reason for proving me wrong. You may just prove under one of the following conditions:
condition 1: pigs can fly but the moon is not made up of cheese.
condition 2: The moon is made up of cheese but pigs cannot fly.
A: Suppose before you roll a 6-sided die, I say "if the die roll is an even number, then it will be a 4." (maybe I'm just guessing).  Next, suppose it comes up a 5.  In that case, surely my predicted claim was not false (since i merely said it would be 4 IF it was even). However, either my prediction was true or it was false (law of the excluded middle), so since we've shown it wasn't false, it must have been true.  In other words, when both the antecedent and the consequent are false, the one-way implication is true.  Finally, a bidirectional implication is true if both of the underlying one-way implications are true, and based on the above, this will hold in cases where both sides are false.
