The truth of P and Q are equivalent The definitions in the right hand column seem almost useless because they are using connectives themselves. Things become much clearer if we form truth tables. T means true and F means false. Q is disturbing to most people at first. It means that you can conclude anything you want from a false proposition.
People tend to read more into a statement like that than just the logical content.
A mathematical example will make you more comfortable. Logical equivalence of A and B means that if we have established that statement A holds, then we immediately know that statement B holds, and vice versa. You can often use truth tables to establish logical equivalence in cases that are not really obvious. We will now show three important examples, each involving the interplay between negation and the other connectives. In the struggle to find a proof of some implication, it is often easier to establish the contrapositive form of the implication and appeal to this logical equivalence to conclude the initial implication is true.