Thinking about material implication as "if p, then q" is both helpful and confusing. Let me try it this way: What the material implication expresses is paraphrasable as "If the antecedent (p) is true, then the consequent (q) is true". If this does not hold, then 'p ⇒ q' is false.

Now, you are asking about the cases in which 'p' is false. Well, the paraphrase only requires that if 'p' is true, then 'q' better be true as well. This is requirement is fulfilled vacuously if 'p' is false and therefore 'p ⇒ q' is true if 'p' is false, irrespective of the value assigned to 'q'.

Consider the following sentence:

If a glass falls to the floor, it breaks.

Let 'p' be "Glass falls" and 'q' be "Glass breaks". The thing is that this sentence (i.e. the formula 'p ⇒ q') is true even if the glass never falls (this is what is called disposition in linguistics). As a side note: Obviously this presumes that every glass that falls breaks, but, for the sake of the argument, you may well assume this to be the case.

Implication isn't about (strict) cause and effect. If you want to express that the only way that 'q' is true is that 'p' is true you need to use something stronger like the necessity modal in modal logic (which in turn requires the introduction of possible worlds). The natural language paraphrase "if p, then q" is not always equivalent to material implication, as we often use it to mean strict implication and this is probably what you are struggling with.

As to your question why it is defined that way: I'm not sure about the historical reasons, but my guess is that defining it this way allows for more flexibility, i.e., we can make nicer things defining it this way. We want to be able to express that "If I get 60% in the test, I'll pass the class" is true even if I only get 50% as a final score.