After navigating the crumbling bridges, the boy takes Corrin through the castle, but it was a trap all along.
Before the Battle
Be sure to have a Locktouch unit in your party. Some enemies do drop keys, but not enough to open all the chests.
Defeat the Boss
In this map, there are areas that have a red floor, and areas that have a blue floor. Enemy units that end their turn on a blue floor will be downgraded into base classes, and enemy units that end their turn on a red floor will be upgraded into advanced classes. So a Ninja that ends its turn on a red floor becomes a Master Ninja, an Archer that ends its turn on a red floor becomes a Sniper, and so on. They will go back to their base classes if they end their turn on a blue floor.
To defeat this mechanism, you could simply stay in the area where you start the battle. Enemies that come through the blue room to reach you will all be downgraded to their base classes.
If you don't have the patience for that, you can make your way into the castle and just fight as you go. There are a couple of Dragon Veins, one on the west side and one on the east side of the map. Using a Dragon Vein will turn the red floors blue and turn the blue floors red. You might want to use a Dragon Vein to turn the floor blue under the boss unit.