No, I can't explain them but I'm willing to speculate.
They supposedly can't be counterfeited and the amount in circulation is tightly regulated. So, they can be made scarce,
Though scarcity is dependent on people desiring them. Initially, I'm guessing, a lot of that was based on novelty and geeks liking the whole idea of being in on something new and stealth like.
And, of course, it's ideal for an underground economy.
So, in a way, bitcoins can be seen as the coinage of a new underground government, the government of a nation that isn't based on geography but rather on a loose alliance of people engaged in dubious transactions.
Currently they are getting away with minting money like this but they have already had to tighten things up, it's harder to make a coin now, And afaik most efforts are now group based. The odds of successfully creating a new coin are a lot lower for individuals. Though the payoff is higher, of course.
So, maybe the bitcoiners can be seen as functionaries of this new government. This government gets its cut in that bitcoiners get to take their work home with them. People are ascribing them value because they can be used in the stealth/dark economy.
Which sort of gives one pause if you consider who is willing to give such a cut in order to be stealthy. Though I suppose if you take into account income taxes and transaction fees like sales tax/VAT then there is room for subsidizing bit coining.
Still, I think it has a lot to do with people wanting to be able to transfer wealth without having government monitoring that.
Naturally, really competent law/code enforcement just laughs at the idea and encourages it so as to burrow inside.
My take is that there's an opportunity here but I can't begin to fathom the risks. I see nothing illegal (but I am not a lawyer) by just bit coining but in today's world there is guilt by electronic association.
P.S. In the old days we used gold and society paid for that privilege. We mined far more than we needed for industry and dentistry. We mined more than we needed for jewelry though who can say how much of it people would wear if it were cheaper.
The point being is that there was overhead. But we needed gold in circulation/backing the printing of money in order to stimulate the economy.
Bit coining could work because the underground economy needs a safe currency and bit coining has no competition and the guys behind it seem trustworthy.
Still, I see it all coming apart if they arrest some guy with bombs and they find he was involved.
Edit: I posted before seeing pcchen's explanation. I would have agreed with his explanation of things though I didn't know about the time stamps and other details.