Let me throw this up...it's my extremely brief explaination of why some of us are able to believe in something we can't prove and it can lead to the appearence or complete irrationality.
First, I'm not in the young earth camp myself, I believe that there can be and is great harmony between Science and the Bible as I've said previously.
With that being said, some of us have had experiences which we cannot explain, science and our doctors cannot explain, our "reason and intellect" cannot explain, our concrete worldview cannot explain, what we "know" to be reality cannot explain. It is these multiple "unexplained" or "unexplainable" and "irrational" experiences which when added together give me at least the knowledge that there is something going on outside of the "reality" I live in. I then tried to align the prepondrance of the "this makes no sense" with different views of the supernatural to see what fit. Which "explanation" of the "unseen" aligned with my experience? Which seemed to have the most "rational" and supported evidence for aligning with "truth"?
Once I came to the conclusion that there "is a God", I then had to take the journey to figure out which of the many "religions" gave the best explaination. Repeatedly I came back to Christianity. The archeology, literary documentation, historicity of the internal and external sources, internal rationality etc. which kept putting it ahead of "other religions". Then it comes down to the core teachings. What is it teaching? How is this different than any other religions teachings, how does it line-up with my personal experiences etc. The concept of Grace is unique to Christianity...it is the only religion which at it's core reinforces what I know is true...if God's standard is perfection...I'm not good enough, nothing that I can do of my own action will allow me to elevate myself to perfection. Every other religion has as it's core the belief that the individual can "work" their way to "heaven". This seemed to at it's core set Christianity apart as different from all the rest. Just one point but a distinct point of reference to anchor to to start.
Then it came down to the person at the core, Jesus. (I know and we can actually argue redaction criticism, early dating, later dating, Pauline thought until the cows come home and keep on until morning...give me this, if I take it that the traditional creeds and theological understanding of who He is is correct and an accurate representation of what He said/taught, then this is what I base my beliefs on...I'll take the creeds/historical understanding at their word). Who did He say that He was? What did He teach? What does this mean for me? Is this someone I can follow? What is the cost?
I guess what I'm trying to say is that I believe that once I came to competely believe from experience that there is more than just what I can see...I had to figure out what it was and what the "bigger truth" was. Once I'd accepted that there was a God, everything else in terms of belief then follows this "truth".
I'm rambling and I know it...
Suffice it to say...for many of us...what you would term as "irrational" and "impossible to prove" is just as real to us as the keyboards in front of us which we are typing on.
I don't think that this belief excuses intellectual laziness and/or complely ignoring "reality" but...I'd caution that the human mind has the amazing ability to see what it wants to see, ignore what it doesn't want to see and convince itself of a great many things to hold as "truth" which are demonstrably false...this is true on both sides of the "what I believe" debate.