On the simplest level, the answer is because every proposition in sentential logic ("propositional calculus") must be either true or false. And this appears to be the best possible resolution given the options of True or False.
To state it in more detail, sentential logic normally follows the following three laws (Aristotle was possibly the first to state them):
- Law of identity - any term we use keeps the same meaning in the context of our argument (A= A).
- Law of non-contradiction - we do not accept contradictory claims as both being compossible. (not (A and not A)).
- Law of the Excluded middle - for every proposition (sentence), it must be either true or false (A or not A)
Given these three laws as our foundation, we can come up with operators that work based on the truth or falsity of multiple sentences. Generally, people use "and" , "or" , "xor" , "if, then", "biconditionality" etc.
For the conditional, the question we're asking about the truth of the sentence refers to the whole. For many accounts of sentential logic, we are asking, "does this conditional obtain in the real world"? Let's say the two terms are A and B.
We consider this FALSE when we have A but not B. i.e. the claim that "if it rains, then it pours." for it to be true, it would require us to be able to confirm that every time it rains, it does in fact pour. But if ever it does rain, but it does not pour (A = true, B = false), then we can say this claim is false about the world.
Similarly, we consider the statement to be TRUE about the world when both the antecedent and consequent do occur. e.g., If I am wearing a blue shirt, I am wearing a shirt. Moreover, it always obtains that when I wear a blue shirt, I am wearing a shirt.
Turning to cases with FALSE antecedents, remember that we must decide that they are either true or false. One option would be to give up -- i.e. to change one of the three laws. Doing so, however, takes us outside of classical logic.
The other option is to realize something about what we are saying.
First off, we are not saying A causes B. Instead, we are merely saying that whenever A occurs, B also occurs. In other words, "if/then" does not make a causal claim about the world. Instead, it makes a factual claim about the world where we can even say it backwards and still be right. E.g., if am extending my umbrella, then it is raining.
Given this, think about the false antecedent claims differently. If the question is "have I found a case where A happens and B does not?" If I have not, then I have not disproved whenever A, then B. Thus, I have no reason to believe this is false. To put it another way, a conditional with a false antecedent is not up for a test. I do not know that it does not obtain in the world. Thus, there's no reason for me to say it is false.
Given then that we have only the choices of TRUE and FALSE, I should consider it TRUE.
All of that to say, however, that this is one of the most common worries people have about sentential logic. I'd say it's a point where one needs to recognize a limitation in the tool set rather than think the toolset is broken for what it does. The material conditional in sentential logic is not identical the English construction "if, then" in scope or function.