An Exploration bonus is a moderate amount of bonus XP the player gets for exploring the map. You can only earn it the first time you pass through a place that would award the bonus. Subsequent raids do not give Exploration bonuses for re-treading the same ground.
Typically, but not always, it involves visiting side pathways and other areas that wouldn't qualify as the map's main thoroughfare. The trails on Customs or the tunnels on Factory are good examples. Certain rooms will award it too, like the office room in Factory or the decrepit train station on Woods.
If you have an idea of what parts of the map will award an exploration bonus then you can visit these spots and be awarded these experience points quickly, but because they can only be awarded once per wipe, there isn't a strong incentive to do it. Your character will be able to earn it through standard play, it isn't necessary to drastically change up your playstyle to achieve these bonuses.