Zones are clusters of area panels on the world map which are grouped together by a blue line. In order to "free" the zone you must defeat the zone boss which is the battle tile with the flashing red dot. You do not need to win every battle tile, simply getting to and beating the Zone boss will free the zone.
Zone bosses are slightly more difficult than regular tile bosses with higher levels and more HP. Taking advantage of elemental weaknesses and/or login bonuses will be helpful in taking them down. Once you beat the Zone Boss, that monster card will be added in as a random summon chance to obtain when doing a regular summon.
This trophy will unlock when you have beaten your first Zone Boss.