Biblical inerrancy is a vital doctrine in the Christian faith. It's not only important to believe because it's true, but also because without it, we can't know much else of what is true.
First, it is vital to establish that it is true. The truth of Biblical inerrancy is a necessary extension of its ultimate author. Although the Bible was written by human hands, it is important to note that "All Scripture is given by inspiration of God..." (literally "God breathed", 2 Timothy 3:16a), and that it came about when "...holy men of God spoke as they were moved by the Holy Spirit." (2 Peter 1:21b). In other words, the Bible was ultimately written by God, with its human authors being a proxy. This is what it means when we call it "the word of God".
Being a product of God Himself, the Bible naturally reflects the character of its author. I once saw a meme about an author's struggle to write a character who is smarter than them. As an amateur author myself, I can confirm the struggle. Let's suppose I wanted to add a doctor to my story, for example. It would be quite obvious, especially to a medical professional, that I just don't have the expertise.
But the thing about God is that He does. We are talking about the Holy Creator of the entire universe. He created and maintains everything in the heavens and the Earth. He knows the past, He knows the future, every single second of it is in His control. And He doesn't lie, either.
This is why the Psalmist can tell us "the words of the Lord are pure words". There isn't a single flaw in the Bible, because there isn't a single flaw in the God who inspired it.
So important is this doctrine that without it, we actually have nothing. Without Biblical inerrancy, we have no authoritative source of doctrine. The Bible becomes an unreliable book, and we have no way to figure out which parts we ought to believe and which parts are just a man made mess. We can't say "every word of God is pure" if we haven't got a clue which words are God's. We can't say "the instruction of the Lord is perfect" if we don't know what the Lord has actually instructed us. We can't say "the testimony of the Lord is trustworthy" if the testimony of the Lord is actually just a collection of man's best guesses from 2,000 years ago.
If you're familiar with American politics, think of CNN. CNN advertises itself as "The most trusted name in news". But it's actually a well known fake news company. Far from being the most trusted name in news, there aren't many people who do trust it. Why? Because they make so many "mistakes" (and don't tend to retract no matter how often they get caught) that there's just no hope left for them. The slime trail of a slug is more likely to tell you something true than CNN. Now imagine if the Bible was the same way. Would you trust it?
Let's put it another way: do you trust it? Many people answer "no". And I'm not even talking entirely about atheists or other unbelievers. I'm talking about people who identify as Christians. Nevertheless, when confronted with the clear teaching of Scripture, they will dismiss those Scriptures. How often have you heard the words "Jesus didn't say that", for example? My guess is quite a few, because it's so unbelievably common for a more liberal Christian to assume that, since Jesus does not personally say a specific thing, therefore any verse that does is irrelevant.
But let's take a look at one thing Jesus did say: "If I have told you earthly things and you do not believe, how will you believe if I tell you heavenly things?" (John 3:12). Put simply, if you don't believe everything God says, how will you believe anything God says? If you don't believe in the first creation, how can you believe in the second? If you don't believe in Hell, how can you believe in Heaven? If you do not believe in sin, how will you believe in salvation? The same God, with the same authority, teaches all of these things, and therefore you have no reason to believe anything He says if you do not believe everything He says.
No one takes the word of a fool because he is foolish. No one takes the word of a liar because he is dishonest. But God is neither a liar, nor is He a fool, and His word has proven its claims to inerrancy time and time again. As Christians, we must affirm the doctrine of inerrancy, especially in these troublesome times. We cannot compromise, we cannot waver, we cannot buckle. May the Lord Jesus Christ be glorified by our meticulous study of His infallible Scriptures.