From what I gather, and certainly someone may come in with a much better explanation or refutation is that Bitcoin is in a sense a finite resource (let's just look at the mechanics right now) and if appetite for Bitcoins increases you can't make more of them.
If a central bank is inflating a currency, it means each year that more of that currency exists than it did previously, so people can effectively bid for resources with more currency which increases the relative price. A pack of gum used to cost $.05. Now it's like $2.99 (or whatever). This is primarily due to inflation as far as I can tell.
Likewise the same is true for Bitcoin, except that it has an additional feature in that it's transmutable into a pack of gum, or a share of JPMorgan stock, or a house, or a car, or some service. It's also transferrable - you can move it (again let's ignore how well it works right now) from place to place without having to actually ship it anywhere.
So because Bitcoin is transmutable like a Euro or Dollar it can do similar things, except the difference is that more Euros and Dollars are effectively "mined" and created, whereas there's a limit to how many Bitcoins can ever exist. So as more and more Euros and Dollars are created, you can spend more of those to buy the same Bitcoin.
Granted, this is very much at face value and some of the mechanics aren't quite there - it's actually very expensive right now to move Bitcoin around and it appears to make for a pretty bad currency, but I think this should at least touch on the mechanics or logic.
tl;dr it's like gold.