Ok, basically Zone Battle is an offshoot of the Zone mode, where you fly your craft around a track while continuously getting faster, until you eventually crash. Zone Battle is similar in that you're constantly getting faster, but you also are able to essentially store up boost by running over speed pads. Running over speed pads fills your bar at the bottom of the screen, which acts sort of like a weapon pickup: if you hit square, you'll use the boost and jump forward that many "zones," meaning you'll speed up. If you hit circle, you'll absorb the boost, healing your ship and giving you a momentary shield.
That aspect is important because when you do hit square and use your boost, you'll also leave behind a barrier that you or other racers can hit; hitting the barrier deals damage. The objective is to get to the required zone (i.e., get fast enough) before anyone else.
