In classical logic, why is $(p\Rightarrow q)$ True if $p$ is False and $q$ is True?
If you don't put any money into the soda-pop machine, and it gives you a bottle of soda anyway, do you have grounds for complaint? Has it violated the principle, "if you put money in, then a soda comes out"? I wouldn't think you have grounds for complaint. If the machine gives a soda to every passerby, then it is still obeying the principle that if one puts money in, one gets a soda out.
Similarly, the only grounds for complaint against $p\to q$ is the situation where $p$ is true, but $q$ is false. This is why the only F entry in the truth table occurs in this row.
If you imagine putting an F on the row to which you refer, the truth table becomes the same as what you would expect for $p\iff q$, but we don't expect that "if p, then q" has the same meaning as "p if and only if q".
$p\Rightarrow q$ is an assertion that says something about situations where $p$ is true, namely that if we find ourselves in a world where $p$ is true, then $q$ will be true (or otherwise $p\Rightarrow q$ lied to us).
However, if we find ourselves in a world where $p$ is false, then it turns out that $p\Rightarrow q$ did not actually promise us anything. Therefore it can't possibly have lied to us -- you could complain about it being irrelevant in that situation, but that doesn't make it false. It has delivered everything it promised, because it turned out that it actually promised nothing.
As an everyday example, it is true that "If John jumps into a lake, then John will get wet". The truth of this is not affected by the fact that there are other ways to get wet. If, on investigating, we discover that John didn't jump in to the lake, but merely stood in the rain and now is wet, that doesn't mean that it is no longer true that people who jump into lakes get wet.
However, one should note that these arguments are ultimately not the reason why $\Rightarrow$ has the truth table it has. The real reason is because that truth table is the definition of $\Rightarrow$. Expressing $p\Rightarrow q$ as "If $p$, then $q$" is not a definition of $\Rightarrow$, but an explanation of how the words "if" and "then" are used by mathematicians, given that one already knows how $\Rightarrow$ works. The intuitive explanations are supposed to convince you (or not) that it is reasonable to use those two English words to speak about logical implication, not that logical implication ought to work that way in the first place.
From the other answers, the most convincing and reasonable explanation why logic implication is defined the way it is, is the idea of sufficient (versus "necessary") condition for something to be true.
NOTE: I don't buy the answer/argument that "if-then" is not an equivalent definition of "$\implies$"; it is just that we tend to have a different notion of "if-then" in everyday life, that of a necessary one.
In short:
Any if-then statement that would "break" only when the conclusion is false while the condition is true is a logic implication.
Examples from everyday life
The following are examples of statements that are logical implications.
"if (it is raining), then (there are clouds in the sky)"
- $T$ - $T$ (OK checks out) $T$
- $T$ - $F$ (Oops!) $F$
- $F$ - $T$ (OK, there are clouds in the sky but it doesn't rain, but that does not "break the rule") $T$
- $F$ - $F$ (no rain, no clouds, still a valid statement as far as I can tell) $T$
"if (I find my room not the way I left it), then (someone was in my room)"
- $T$ - $T$ (ok checks out so far) $T$
- $T$ - $F$ ("rule" breaks) F
- $F$ - $T$ (does not break the logic of the "rule") $T$
- $F$ - $F$ (this does not break the logic either) $T$
A more elaborate explanation with example from science
Consider an example in the field of medical diagnosis. The basic (and ideal) premise of diagnosis from symptoms is to derive valid, sufficient rules that can safely conclude a diagnosis of an illness over other illnesses based on symptom observations. Let's say some medical scientist studies illness A and proposes the following diagnostic rule:
"IF (symptom B and symptom C are observed) THEN (--for sure-- the patient is under illness A)."
He then goes through all the documented cases of the illness (or conducts a new study) and tries to see whether that rule is valid:
If a patient in the records had the symptoms and he was also found to have illness A (1st row of the truth table), then so far so good.
If a patient is found to have the symptoms but not the illness, that breaks or falsifies the rule (2nd row in the truth table), and the rule has to be reconsidered and revised because it simply does not work; the rule, as a logic implication, is false.
If some patient is found to have illness C but not the symptoms (3rd row in the truth table), that does not reduce the validity of the rule in any way as a way of making a safe conclusion; it only reduces its usefulness, depending on how many cases it may miss.
If some patient is found not to have the symptoms nor the illness (4th row in the truth table), that is irrelevant to the validity of the rule.
So, if the scientist finds only records of the 1st, 3rd, and 4th case, then he has a valid rule. Moreover, the potential of the rule to break in the 2nd case makes it a logic implication.
From the above, you can see that the way logic implication is defined (with the third and fourth rows being True) finds extensive use in math and science and, eventually, is what makes sense.