I'm fortunate to have a Master's degree in Christian Apologetics from a seminary
with a strong apologetics focus, having been taught my apologetics and philosophy by one of the top Christian defenders alive today — (Dr. Norman Geisler
). This means I am well-versed in the defense of the Bible and can provide various evidences (e.g. historical, archaeological, philosophical, prophetic, etc.) as to why the Bible is trustworthy and should be believed.
I also have a Ph.D. in New Testament with my dissertation being on the apologetics of the Apostle Paul. This means I can carry on conversations about manuscript evidence, the internal consistency of the New Testament and much more, all of which add extra weight as to why the Bible should be believed.
Even though I am schooled in all these things, and value the information greatly, they are not why I believe the Bible is true.
Skeptics and the Bible
Those who distrust all or parts of the Bible do so for numerous reasons. For many, the primary cause is a presuppositional bias against the supernatural. A man coming back from the dead, walking on water, instantly curing diseases and more immediately raise red flags to the skeptic because such things are not routinely observed and therefore are dismissed as fable.
For others, the contents of Scripture simply rub them the wrong way. The God of the Old Testament seems merciless and cruel, the moral pronouncements seem outdated and go against their chosen desires/lifestyle, or they don't like the idea of Hell.
Still others object to the Bible on philosophical grounds. They don't understand, for example, how an all-powerful and loving God could permit evil to occur in the world.
Although these are all questions/issues that deserve good answers (which have been given), these explanations are not why non-Christians don't believe the Bible to be true.
Why People Embrace the Bible
If you're a Christian, do you remember when or how you came to believe in the truthfulness of the Bible? I do.
I was an engineering/business student in college. We went to church, but I couldn't have cared less. I'd never opened the Bible myself one time in my life and read it.
Then, for reasons I still can't explain, I bought a book on Bible prophecy. Maybe it was because of all the math and statistics I was taking in school — I was amazed at the Bible's predictions and the impossibility of those prophecies occurring or being faked.
For the first time I also realized that if Jesus really did come back, I was in big trouble. At the end of the book was a prayer on how to receive Christ, which I did.
Everything changed for me then, especially where the Bible was concerned.
I didn't know what apologetics was and couldn't make a defense of the Bible to save my life, but when I first started reading the Bible for myself, I knew what I was reading was the truth.
Such is the case for most Christians. Maybe God brought them to Himself through existential circumstances or via a more cerebral route. As C. S. Lewis observed: "Nearly everyone I know who has embraced Christianity in adult life has been influenced by what seemed to him to be at least a probable argument for theism." 
It's not that we believe for no reason. There is always some instrumental cause God uses to sink His truth into us whether it's through the heart or head. But make no mistake about it: if you're a Christian and believe the Bible, it's not because you're smarter (or dumber by the skeptic's way of thinking...), figured things out where other's haven't, or are better in some way than any other person.
You believe the Bible because God loved you, saved you, gifted you with the Holy Spirit, and by grace opened your eyes to believe His Word.
One particular verse in Scripture (on which I built my entire 300-page doctoral dissertation) spells this out beautifully: "For we know, brothers loved by God, that he has chosen you, because our gospel came to you not only in word, but also in power and in the Holy Spirit and with full conviction" (1 Thessalonians 1:4–5).
Why did Paul's listeners in Thessalonica believe the truth of God that Paul brought them while others rejected the apostle's preaching? Because God loved them, selected them to be His children, and gifted them with the Spirit so they believed with true conviction.
This is why anyone believes the Bible. Scripture is clear on this point and also exposes why God chooses this method: "For consider your calling, brethren, that there were not many wise according to the flesh, not many mighty, not many noble; but God has chosen the foolish things of the world to shame the wise, and God has chosen the weak things of the world to shame the things which are strong, and the base things of the world and the despised God has chosen, the things that are not, so that He may nullify the things that are, so that no man may boast before God" (1 Corinthians 1:26–29).
You believe the Bible because God specifically called you out of darkness and in the end He gets all the credit for it.
Why People Reject the Bible
What about those who don't accept the Bible? Scripture is equally plain as to why this is the case:
• "Jesus said, 'I praise You, Father, Lord of heaven and earth, that You have hidden these things from the wise and intelligent and have revealed them to infants. Yes, Father, for this way was well-pleasing in Your sight. All things have been handed over to Me by My Father; and no one knows the Son except the Father; nor does anyone know the Father except the Son, and anyone to whom the Son wills to reveal Him,'" (Matthew 11:25–27).
• "To you it has been given to know the secrets of the kingdom of heaven, but to them it has not been given...This is why I speak to them in parables, because seeing they do not see, and hearing they do not hear, nor do they understand," (Matthew 13:11–15).
• "And Jesus said to him, 'Blessed are you, Simon Barjona, because flesh and blood did not reveal this [that Jesus is the Christ] to you, but My Father who is in heaven,'" (Matthew 16:17)
• "Why do you not understand what I am saying? It is because you cannot hear my word. You are of your father the devil," (John 8:43–44).
• "Whoever is of God hears the words of God. The reason why you do not hear them is that you are not of God," (John 8:47).
• "You do not believe because you are of My sheep" (John 10:26).
• "For the mind that is set on the flesh is hostile to God, for it does not submit to God's law; indeed, it cannot," (Romans 8:7).
• "But a natural man does not accept the things of the Spirit of God, for they are foolishness to him; and he cannot understand them, because they are spiritually appraised," (1 Corinthians 2:14).
• "Even if our gospel is veiled, it is veiled to those who are perishing," (2 Corinthians 4:3).
• "We are from God. Whoever knows God listens to us; whoever is not from God does not listen to us. By this we know the Spirit of truth and the spirit of error" (1 John 4:6).
While non-Christians may provide many explanations as to why they discard the Bible, the actual reason they don't believe is that God hasn't spiritually awakened them (yet). The objection of not having enough proof fails to hold water because (1) there is plenty of evidence that attests to the Bible's validity; (2) people act contrary to evidence all the time, as I point out in my article
that discusses evidence and hard core unbelief.
To Whom was the Bible Written?
Stating what few modern day preachers would dare say or put in print today, A. W. Tozer says:
The Bible is not addressed to just anybody. Its message is directed to a chosen few...Some believe and some do not; some are morally receptive and some are not; some have spiritual capacity and some have not. It is to those who do and are and have that the Bible is addressed. Those who do not and are not and have not will read it in vain...As the pillar of fire gave light to Israel but was cloud and darkness to the Egyptians, so our Lord's words shine in the hearts of His people but leave the self-confident unbeliever in the obscurity of moral night." 
While theology can be learned by anyone, the acceptance and adherence to spiritual truths such as those contained in the Bible cannot be done without being born again (John 3). This being the case, as Christians we should thank God for opening our previously blind spiritual eyes and hardened hearts to His Word and pray for those who do not embrace the Bible yet and ask God to have mercy on them just as He did on us (2 Timothy 2:25).
1. C.S. Lewis; Case for the Christian Faith
2. A. W. Tozer, "Why People Find the Bible Difficult" in The Best of A. W. Tozer
, ed. Warren W. Wiersbe (Camp Hill, PA: Baker, 1980), pp. 164-166.