Upon entering the Eastern Palace, head north through the first few rooms, defeating Popos as you go. You'll soon reach a room with a large gap and a moving platform. Use your merge ability to cross the gap by merging onto the wall and moving along it as the platform approaches. In the next room, you'll find a switch that activates a bridge. Hit it with your sword or an item.
Continue through the dungeon, solving simple push-block puzzles and using the Bow to activate distant switches or defeat Armos. Remember to look for cracks in walls; these often indicate bombable sections leading to secrets or shortcuts. The dungeon's layout encourages exploration, so don't be afraid to backtrack if you feel stuck. Eventually, you'll find the Big Key, which is usually guarded by a mini-boss or a more complex puzzle. Once you have it, locate the large boss door and prepare for the encounter.