Substantially means "to a great extent." Your college fund was substantially increased by your grandfather's generous gift. What the heck, I'll throw in a quarter, too.
Substantially is a good word to know any time you want to establish something's extent. It basically suggests "a large part." Your raise made your life substantially easier. The word also suggests an essential nature, as in "Nothing was substantially changed by the new law." The word carries a great deal of weight, as can be seen in this quote by Albert Einstein: "We shall require a substantially new manner of thinking if mankind is to survive."