Being falsifiable is what makes something open to being testable, and therefore being capable of standing as evidence. If something doesn't allow for the possibility it is wrong, it cannot be tested, and can't stand as evidence. If I say to you that I have a unicorn in my closet but you cannot test it, you cannot question, and you're not allowed to see it... How foolish would you be to believe it? Very, right?
Falsifiability is a huge part of being able to form beliefs or opinions. If you have 3 things for example that you can't test, that are unfalsifiable, how can you determine which of the 3 are correct? For example, Christianity, Islam, and Judaism. None allow for the possibility of being wrong or in any part inaccurate. So how can you determine which is true? You can't, because all are different, and none allow for the possibility that they are wrong. There's no way to test it, there's no way to compare them.
Provable pretty much goes in hand with it. If I can't prove to you that I have said unicorn and you can't question or observe the claim or any part of it, how can you accept it? Even if I offer "proof" for example, a DNA sample but will not allow you to test it or claim that any results are wrong if they counter what I claim, how can you verify it? This is essentially what holy books do. No proof to support them, and anything that counters them simply must be wrong even if it's right. Especially when overwhelming evidence suggests my claim is impossible in addition to my not allowing it to be tested. A massive chunk of the bible is wrong. The entire creation account is absolutely disproven for example. There is no question, there is no doubt, it did not happen that way. Period. Mapping of genetics alone proved it, but we have countless other things that do so as well.Very few positive claims in the bible are falsifiable, cannot be tested or observed, no trace of them in history or science. The few that can be, are not unique to the bible or were already well known at their time (thus not rendering them as acceptable to support the claims).
Long story short, if you can't show it, you don't know it. If you don't allow for it to be tested, it can never have it's veracity demonstrated and is therefore by default dismissed.