Extraordinarily means to an exceptional degree, far beyond what's usual or ordinary. When something happens extraordinarily well, it exceeds normal expectations by a huge margin.

If you score well on a test, that's good. If you score extraordinarily well, you might get a perfect score and solve bonus problems that stumped everyone else. An extraordinarily talented musician plays with such skill and feeling that audiences are amazed. An extraordinarily difficult puzzle is so challenging that even experts struggle with it.

You might say a gymnast performed extraordinarily, that someone was extraordinarily kind during a difficult situation, or that a storm was extraordinarily powerful.

Notice that extraordinarily can describe both positive and negative extremes. A day can be extraordinarily beautiful or extraordinarily awful. Either way, it's standing way out from the everyday middle ground where most things happen.