Alright, here we go:
Octane means the fuel's ability to resist detonation. There are certain times when you want the fuel to detonate; and other times you don't.
Increased compression; whether built in or added by a supercharger, causes heat. Heat is what ignites the fuel.
You don't want the fuel to detonate before the spark ignites; if this happens, the flame front from the compression ignited fuel will collide with the flame front from the spark ignited fuel. Now whether than having a nice slow burn, you now have a violent explosion!
That violent explosion (commonly referred to as pre-detonation or ping) is what breaks piston rings and punches holes in the top of pistons.
Broken piston rings will cause heavy oil consumption and a huge plume of blue smoke emitting from your tail while driving down the freeway.