When I play dwarf fortress, at the map generation stage when I'm deciding where to settle, I can see a huge expansive map with a lot of places to explore. However, once settled, it doesn't take me long before I reach the edge of this new smaller area. Is it possible to venture out to the bigger world, or is that now the new edge of the world?
When you embark on a new fortress, you have the option of choosing the size of the embarkment area for your new fortress. This size, which usually defaults to 4x4, is the size on the large world map which your fortress will occupy. Outside this area, the rest of the world still exists - migrants1, diplomats, nobles and goblin sieges all must come from valid civilizations that exists elsewhere. History still moves on, and occasionally you get hints of that, such as when Ages change.
From your perspective, however, the new playing field will be all that you have - there is no way of commanding dwarfs to move outside this zone, nor to expand the zone once you start a game. You will have to make do with the resources you can find in this area. To get more, you can trade with caravans that comes once in a while, or strip them off dead enemies that attack your fortress. Anything that moves out of the zone, like liquids that flow outside the zone, items stolen by animals, children kidnapped by Kobold thief, etc., will be considered outside the simulation as far as the game is concerned.
You might think this zone is small, but remember that the world in Dwarf Fortress is three dimensional. The world extends hundreds of levels downwards, and you should have no shortage of space once you start excavating. To really see the "world", you have to play Dwarf Fortress in Adventurer mode, though this is not possible while there is a Fortress still being played in the same world.
1 The only exception here is that first two waves of migrants are coded such that they will always arrive, regardless of the size of your parent civillization.