Good question! I can give you the answer in three words:
Because. It. Works.
If being a Christian didn't produce real, verifiable empirical results, I would not waste years of my life following the teachings of Jesus. But from my point of view—which is as objective as I can make it—the results, effects, ramifications and manifestations of my particular faith all testify to the accuracy of the testimonies of scripture.
Christian doctrine says that as a believer, I am supposed to have the ability to change the things about myself that are hateful and obnoxious, and to eradicate the "old nature" inherited from my ancestors, and put on the "new nature" of Jesus by faith and obedience to the scriptures so I can live a life that's free from the constraints of sin. And my religion works exactly like it's supposed to: it enables me to do just that.
Christian doctrine says that God answers prayer (which sometimes means waiting till God determines the perfect time to do it, rather than getting everything the instant I ask for it), and that works exactly like it should. Christian doctrine says God puts a new heart and a new spirit inside us, gives us peace that is beyond human reason, and puts love in our hearts for our fellow creatures: and that has worked exactly right, too.
Christian doctrine declares that the holy scriptures are "spirit and life" and that they provide wisdom, discretion, understanding, and help our faith to increase if we commit them to memory and hide them in our hearts by meditating on them; and that works for me too. Everything promised to believers in the Bible, and every piece of evidence the Bible offers by which a person can prove and confirm the operation of God's Holy Spirit in his or her life, are all true and accurate according to my experience. The truth is supposed to work, isn't it?
If your personal experience doesn't prove to you that something works, you sure aren't going to accept any less significant form of evidence, are you? Experience is a great teacher. And once you perceive by experience that what you believe to be the truth is in fact the truth...then what? You have to respect the authority of the person who is the author of that truth, and accept the entire body of his work.
So when Jesus says "I am the way, the truth, and the life" and then insists that there is no other way a person can have access to God but by him, I take his word for it. I hold it to be absolutely true because everything else he said—which I have effectively proved, over time, to my own satisfaction by my own personal experiences—has worked exactly as he said it would. That's the reason why I'm sure my religion is true to the exclusion of all others.
Have I considered that it may all be wrong? That would be kind of ludicrous. If something works for me just like it's supposed to, why should I reject it on grounds that there are lots of counterfeits floating around? Besides, my religion is not a set of rules, or rituals, or constant striving to attain a state of divinity. My religion consists of having personal fellowship with the Father of mankind, and with his appointed savior and son, Jesus Christ.