Something that happens extensively occurs in a wide or broad way — on a large scale or across a large area. When a hurricane extensively affects a state, the damage is spread over many miles.
You might decide to extensively edit your anthropology paper after reading the one your friend wrote, or brag that your band will be touring extensively this summer, covering the entire Midwest. In either case, your effort is on a large scale. Extensively comes from the adjective extensive, and both words are rooted in the Latin extendere, "stretch out," from ex, "out," and tendere, "to stretch."