After deciding to defend Hoshido, you must battle the Nohr.
Defeat the Boss
The victory condition is to defeat the boss. In this battle, Hinoka, Takumi, and Ryoma won't gain EXP, but Sakura and your other units will.
Have Corrin and another unit stand next to each other to fight the nearby enemy together. Repeat this process to defeat Leo after he comes closer.
In Normal and Hard mode, Xander only attacks Ryoma if Ryoma is in his range. To prevent Ryoma from dying, you can move him next to Corrin or another of your units on the bridge. Xander will come attack Ryoma during the Enemy Phase.
If Xander hasn't been defeated yet, you can go after Elise and Camilla for extra EXP.
After the battle, Lilith takes you to your castle in the astral plane and walks you through creating buildings. At this point you can have Support conversations with your units, invite an ally to your private quarters (be sure to select someone with a heart icon next to the name because this will ensure that your relationship will improve), collect resources, and buy stuff from the shops.
Be sure to feed Lilith. You might want to check on the crystal ball to visit other castles (if you are connected to the Internet) because this will let you gather resources that are unavailable in your own castle, giving you a wider range foods to give stat boosts to Lilith.
When you are ready, press Start to go to the castle menu, then choose Next Battle. There will be several options. The Dragon's Gate lets you purchase DLC (be sure to check for free DLC). You can also change the difficulty here, as well as do some random Challenge battles, or you can spend gold to do Scout battles, which will expire if you don't finish them before the next chapter.
To continue, choose the Chapter 7 option from the castle menu. The game will progress to Chapter 7: A Vow Upheld.