In The Voyage of the Dawn Treader, exploration is the driving force behind everything our beloved protagonists do. The story is explicitly structured as a quest, and discovery for its own sake is just as important as any more practical goal (such as finding the lost lords of Narnia). Exploration is almost synonymous with adventure in this book; exploring the world seems to be inherently dangerous but also thrilling and rewarding. There are larger forces at work (think Aslan) in the world of this book that protect explorers and ensure that their travels end well.
The Dawn Treader voyages to the east because that is the direction of the sunrise and symbolically represents rebirth.
Although Caspian is officially searching for the seven missing lords, in reality he is searching for personal fulfillment and transformation.