93 octane is more explosive than 95. In a high compression engine (typically turbo or supercharged), this can cause pre-ignition of the fuel-air mixture, which is obviously not very healthy. Modern engines use knock-sensors to detect this, and automatically retard the timing to compensate, but this can reduce the power of the engine.
So you put a higher octane fuel in, and because the pre-ignition isn't happening, the engine management doesn't have to retard the timing, and the engine can put out the correct power.
When it comes to lower compression engines that don't have pre-ignition issues, there's no difference between high and low octane fuel, because the engine doesn't need timing changes on either.
