In each of the 5 worlds there is an "explore" icon before the main story battles. In Innistrad it will say "explore Innistrad" and in ravnica it will say "explore Ravnica", and so on. Each has a set of 5 battles set in that world that are not tied to the story in any way. You can not click on the first explore icon until you have defeated the Avacyn battle in Innistrad. Once you have defeated her you can choose to explore each world as soon as you unlock it.
Each of these icons have 5 different battles in a random order and no battle will repeat until you have won all 5. You will have to defeat a total of 25 battles to win this achievement (5 battles each in all 5 worlds)
If you have not purchased the card collection in the marketplace, after the 5th battle you win in each world you will receive a special booster pack containing 4 additional rares. The player stats page will tell you how many worlds you have fully explored but not tell you which they are. So the easiest way to do this is to battle the same world until the stats page counter goes up by one and then move on to the next one.
These explore matches are the only way to continue to unlock booster packs after you have defeated Garruk in the main story.