Premium gasoline (high-octane gasoline, actually) is harder to ignite than regular gasoline. This means it is less likely to ignite from the heat or pressure of the engine, and will only do so when it gets a spark from the spark plug. This means that the engine can run at higher pressures and/or temperatures.

You have a high-compression engines which is designed to run at those high pressures and temperatures. That is why it calls for premium gasoline.

If you run low-octane gas in a high-octane engine, what can happen is the fuel will ignite before it is supposed to. Because the piston won't be in the right position when this happens, it will be slammed backwards (or at least slowed down, with a lot of stress on the bearings and whatnot) for an instant before continuing on it's path, causing what's called "knock."

Now your car is designed to cope with this. It can alter the timing to prevent knock. But the typical cost of this is less power and poor fuel economy. And there's no guarantee it will work perfectly forever...particularly as the car gets older. Carbon buildup in the engine can increase heat causing preignition. In a normal car, if this (or anything else that causes knock) happens, you can switch to premium gas and get a few more years out of the engine before you have to do anything serious about it. If this happens in an engine meant to run premium gas...well, you'll have no off-the-shelf options to go to from there.

The car's ability to cope with low-octane gasoline is meant to allow you to keep driving if you find yourself someplace where you can't get premium gas. It isn't meant to save you money on gas.

Basically, any money you save not buying high-octane gas, you'll lose in either lost fuel economy, or early engine rebuild.