That's not the value of the bitcoin, it's the value of the cryptographic code which has been open sourced. Anybody can fork it and create its own "Webcoin", but there's no intrinsic value in any forked instance coin, beyond the publicity that allows early adopters to cash in.

So their unique value comes from private transactions...but there's only a finite supply of bitcoins? How many of these transactions can actually occur in 2013-2014, when every bitcoin has been mined and their value is skyrocketing?

The amount of transactions that can take place is unlimited, and even though there is a maximum of 2.1 million coins, each coin is divisible to 8 decimal places, giving 2.1 quadrillion units of currency.

Also, mining won't be completed this century, although the rate at which coins are generated decreases over time.

Cash, delivered in person, does the same thing.

Which has an obvious disadvantage over bitcoins: it has to be delivered in person.