Why is the statement p"implies"q true if p is false? M: The truth table of p→q says that if p is false and then p→q is true.
The author of the books goes on to clarify the doubt people have about the above statement, he uses the following scenario to explain why M is true, 
"Perhaps you are bothered by the fact that P ⇒ Q is true in the last two
lines of this table. Here’s an example that should convince you that the
table is correct. Suppose you have just taken a test, and you ask your
professor if it will be graded by the next class. Your professor makes the
following promise:
If you come to class, then you will get your test.
Your professor is making the promise
(You come to class) ⇒ (You get your test)."
He goes on to say that if you don't come to class and (get your paper)/(do not get your paper), then he din't lie, if you don't lie you say the truth, therefore he din't lie. But how can we be sure he dint lie?, the promise wasn't broken, sure, but we are not sure that it wouldn't have been broken if we had previously decided to go to class instead of not going.
ps; I haven't taken any logic courses.
 A: $p \Rightarrow q$ by definition means $\lnot p \lor q$. Everyday language can give an intuition of the reason behind this: "if they don't discount it, I won't buy it" and somewhat equivalently "either they'll discount it or I won't buy it".
A line in a truth table is to be interpreted as a disjunction of conjunctions of elementary propositions. So, if you look at the truth table for implication:
$$\begin{array}{lcccc}
p & 1 & 1 & 0 & 0 \\
q & 1 & 0 & 1 & 0 \\
p \Rightarrow q & 1 & 0 & 1 & 1
\end{array}$$
One way of reading it is "(p and q) or (not p and q) or (not p and not q)", which is equivalent to "q or not p".
In other words, you have to take a holistic reading of the whole line, which is your definition. It doesn't make sense to say, for example, "if p is false, then it is true that p implies q".
A: When we say that a formula (about the real numbers, say) such as "If $x < 5$, then $x < 10$" is true, that means, it is true for ALL values of the variable $x$, including, for example, $x=12$  (F implies F), $x=8$ (F implies T), and $x=2$ (T implies T).
This is indeed a convention of mathematical language/notation, but a very convenient one, because we'd like the interpretation of "Formula $A(x)$ is true" to be the same for implications $A(x)$ such as the one above as for e.g. $A(x)$ being $(x^2-1)=(x+1)(x-1)$, namely that the formula holds for all values of $x$.
