Imagine this statement
If there is sun then it will be a bright day
which is of the form $p \to q$. If the sun was there and it was not bright, then the truth value of this statement would be false. So why did I make this statement instead of keeping quiet? As far as common sense and wisdom go, one would prefer to avoid making such statements if it is not always going to be true.
I think I am still not clear on the usage of such propositions. Is this a bad example from "spoken language" to understand why we need this?
In Mathematics, you start with a statement whose truth value is unknown and then enumerate all the possible scenarios to determine the truth value of $p \to q$ as opposed to "spoken language" where you actually (implicitly) mean it is true without a doubt (if I had doubt, I wouldn't say it)? The key point being I start with an unknown trying to establish the truth value?