I think the bacteria on the rock (hence 'live') effectively filters the water. Though I'm a complete marine noob, someone with a marine tank should be able to answer it in a breeze.
Yeah, marine only I'm pretty sure. If you put live rock in a FW tank the bacteria will be dead on it and pollute the water, it will also release salt into the water and affect the pH/hardiness I believe.
live rock alone can be used as filtration because of all of the little organisms on it. It is extremely helpful when cycling a marine tank because if you do have enough live rock it can cycle your tank in 2 days. as it did with mine. thats the reason it is so expensive.
Try sending a sack of potatoes around the world in 24 hours. That would cos a hell of a lot.
Live rock and flow are also the most important things in most reef tanks
But you can also have 'live rock' in FW set ups. Its a piece of rock, bogwood or gravel from an already established FW setup. This will harbour the good bacteria for cycling a new tank. If you have many tanks in your home you can add some gravel from a established tank. It will make the cycling quicker but not filter.
Do not mix SW live rock with FW it would be a very very expensive mistake!
What reall makes it so expensive is that it has to first be collected by SCUBA divers in relatively remote locations. It then has to be screened and modified so that it passes local regulations. CITES or similar permits must then be secured for the rock at its holding place nearby its point of collection. Then once the permits are secured the rock must be airfreighted away from that remote location (likely in the pacific) halfway around the world. Rock is heavy and air-shipping it isn't exactly cheap . Then it needs to pass inspection and be held at the destination country before sale. Its a long trip to get it from ocean to your tank, hence not quite cheap